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Christoph Schiller MOTION MOUNTAIN the adventure of physics – vol.iv the quantum of change www.motionmountain.net Christoph Schiller Motion Mountain The Adventure of Physics Volume IV The Quantum of Change Edition 31, available as free pdf with films at www.motionmountain.net Editio trigesima prima. Proprietas scriptoris © Chrestophori Schiller primo anno Olympiadis trigesimae secundae. Omnia proprietatis iura reservantur et vindicantur. Imitatio prohibita sine auctoris permissione. Non licet pecuniam expetere pro aliqua, quae partem horum verborum continet; liber pro omnibus semper gratuitus erat et manet. Thirty-first edition. Copyright © 1990–2021 by Christoph Schiller, from the third year of the 24th Olympiad to the first year of the 32nd Olympiad. This pdf file is licensed under the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Germany Licence, whose full text can be found on the website creativecommons.org/licenses/by-nc-nd/3.0/de, with the additional restriction that reproduction, distribution and use, in whole or in part, in any product or service, be it commercial or not, is not allowed without the written consent of the copyright owner. The pdf file was and remains free for everybody to read, store and print for personal use, and to distribute electronically, but only in unmodified form and only at no charge. To Britta, Esther and Justus Aaron τῷ ἐμοὶ δαὶμονι Die Menschen stärken, die Sachen klären. PR E FAC E “ ” Primum movere, deinde docere.* Antiquity T his book series is for anybody who is curious about motion in nature. How do hings, people, animals, images and empty space move? The answer leads Motion Mountain – The Adventure of Physics o many adventures, and this volume presents those due to the discovery that there is a smallest possible change value in nature. This smallest change value, the quantum of action, leads to what is called quantum physics. In the structure of modern physics, shown in Figure 1, quantum physics covers four of eight points. The present volume introduces the foundations of quantum theory, deduces the structure of atoms and explains the appearance of probabilities, wave functions and colours. The present introduction to quantum physics arose from a threefold aim I have pur- sued since 1990: to present the basics of quantum motion in a way that is simple, up to date and captivating. In order to be simple, the text focuses on concepts, while keeping mathematics to the copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net necessary minimum. Understanding the concepts of physics is given precedence over using formulae in calculations. The whole text is within the reach of an undergraduate. In order to be up to date, the text is enriched by the many gems – both theoretical and empirical – that are scattered throughout the scientific literature. In order to be captivating, the text tries to startle the reader as much as possible. Read- ing a book on general physics should be like going to a magic show. We watch, we are astonished, we do not believe our eyes, we think, and finally we understand the trick. When we look at nature, we often have the same experience. Indeed, every page presents at least one surprise or provocation for the reader to think about. Numerous interesting challenges are proposed. The motto of the text, die Menschen stärken, die Sachen klären, a famous statement on pedagogy, translates as: ‘To fortify people, to clarify things.’ Clarifying things – and adhering only to the truth – requires courage, as changing the habits of thought produces fear, often hidden by anger. But by overcoming our fears we grow in strength. And we experience intense and beautiful emotions. All great adventures in life allow this, and exploring motion is one of them. Enjoy it. Christoph Schiller * ‘First move, then teach.’ In modern languages, the mentioned type of moving (the heart) is called motiv- ating; both terms go back to the same Latin root. 8 preface Complete, unified description of motion Adventures: describing precisely all motion, understanding the origin of colours, space -time and particles, enjoying extreme thinking, calculating masses and couplings, catching a further, tiny glimpse of bliss (vol. VI). PHYSICS: An arrow indicates an Describing motion with precision, increase in precision by i.e., using the least action principle. adding a motion limit. Quantum theory General relativity with classical gravity Quantum field theory Adventures: the Adventures: bouncing (the ‘standard model’) neutrons, under- Adventures: building Motion Mountain – The Adventure of Physics night sky, measu- ring curved and standing tree accelerators, under- wobbling space, growth (vol. V). standing quarks, stars, exploring black bombs and the basis of holes and the life, matter & radiation universe, space (vol. V). and time (vol. II). Classical gravity Special relativity Quantum theory Adventures: Adventures: light, Adventures: biology, climbing, skiing, magnetism, length birth, love, death, c contraction, time copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net space travel, chemistry, evolution, the wonders of limits dilation and enjoying colours, art, astronomy and G fast E0 = mc2 h, e, k paradoxes, medicine geology (vol. I). limits motion (vol. II). limit and high-tech business uniform tiny (vol. IV and vol. V). motion motion Galilean physics, heat and electricity The world of everyday motion: human scale, slow and weak. Adventures: sport, music, sailing, cooking, describing beauty and understanding its origin (vol. I); using electricity, light and computers, understanding the brain and people (vol. III). F I G U R E 1 A complete map of physics, the science of motion, as ﬁrst proposed by Matvei Bronshtein (b. 1907 Vinnytsia, d. 1938 Leningrad). The Bronshtein cube starts at the bottom with everyday motion, and shows the connections to the ﬁelds of modern physics. Each connection increases the precision of the description and is due to a limit to motion that is taken into account. The limits are given for uniform motion by the gravitational constant G, for fast motion by the speed of light c, and for tiny motion by the Planck constant h, the elementary charge e and the Boltzmann constant k. preface 9 Using this b o ok Marginal notes refer to bibliographic references, to other pages or to challenge solutions. In the colour edition, marginal notes, pointers to footnotes and links to websites are typeset in green. Over time, links on the internet tend to disappear. Most links can be recovered via www.archive.org, which keeps a copy of old internet pages. In the free pdf edition of this book, available at www.motionmountain.net, all green pointers and links are clickable. The pdf edition also contains all films; they can be watched directly in Adobe Reader. Solutions and hints for challenges are given in the appendix. Challenges are classified as easy (e), standard student level (s), difficult (d) and research level (r). Challenges for which no solution has yet been included in the book are marked (ny). Advice for learners Learning allows us to discover what kind of person we can be. Learning widens know- Motion Mountain – The Adventure of Physics ledge, improves intelligence and provides a sense of achievement. Therefore, learning from a book, especially one about nature, should be efficient and enjoyable. Avoid bad learning methods like the plague! Do not use a marker, a pen or a pencil to highlight or underline text on paper. It is a waste of time, provides false comfort and makes the text unreadable. And do not learn from a screen. In particular, never, ever, learn from the in- ternet, from videos, from games or from a smartphone. Most of the internet, almost all videos and all games are poisons and drugs for the brain. Smartphones are dispensers of drugs that make people addicted and prevent learning. Nobody putting marks on paper or looking at a screen is learning efficiently or is enjoying doing so. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net In my experience as a pupil and teacher, one learning method never failed to trans- form unsuccessful pupils into successful ones: if you read a text for study, summarize every section you read, in your own words and images, aloud. If you are unable to do so, read the section again. Repeat this until you can clearly summarize what you read in your own words and images, aloud. And enjoy the telling aloud! You can do this alone or with friends, in a room or while walking. If you do this with everything you read, you will reduce your learning and reading time significantly; you will enjoy learning from good texts much more and hate bad texts much less. Masters of the method can use it even while listening to a lecture, in a low voice, thus avoiding to ever take notes. Advice for teachers A teacher likes pupils and likes to lead them into exploring the field he or she chose. His or her enthusiasm is the key to job satisfaction. If you are a teacher, before the start of a lesson, picture, feel and tell yourself how you enjoy the topic of the lesson; then picture, feel and tell yourself how you will lead each of your pupils into enjoying that topic as much as you do. Do this exercise consciously, every day. You will minimize trouble in your class and maximize your teaching success. This book is not written with exams in mind; it is written to make teachers and stu- dents understand and enjoy physics, the science of motion. 10 preface Feedback The latest pdf edition of this text is and will remain free to download from the internet. I would be delighted to receive an email from you at fb@motionmountain.net, especially on the following issues: Challenge 1 s — What was unclear and should be improved? — What story, topic, riddle, picture or film did you miss? Also help on the specific points listed on the www.motionmountain.net/help.html web page is welcome. All feedback will be used to improve the next edition. You are welcome to send feedback by mail or by sending in a pdf with added yellow notes, to provide illustrations or photographs, or to contribute to the errata wiki on the website. If you would like to translate a chapter of the book in your language, please let me know. On behalf of all readers, thank you in advance for your input. For a particularly useful contribution you will be mentioned – if you want – in the acknowledgements, receive a Motion Mountain – The Adventure of Physics reward, or both. Support Your donation to the charitable, tax-exempt non-profit organisation that produces, trans- lates and publishes this book series is welcome. For details, see the web page www. motionmountain.net/donation.html. The German tax office checks the proper use of your donation. If you want, your name will be included in the sponsor list. Thank you in advance for your help, on behalf of all readers across the world. The paper edition of this book is available, either in colour or in black and white, copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net from www.amazon.com, in English and in certain other languages. And now, enjoy the reading. C ON T E N T S 7 Preface Using this book 9 • Advice for learners 9 • Advice for teachers 9 • Feedback 10 • Support 10 11 Contents 15 1 Minimum action – quantum theory for poets Motion Mountain – The Adventure of Physics The effects of the quantum of action on rest 19 • The consequences of the quantum of action for objects 20 • Why ‘quantum’? 22 • The effect of the quantum of action on motion 24 • The surprises of the quantum of action 26 • Transformation, life and Democritus 28 • Randomness – a consequence of the quantum of action 32 • Waves – a consequence of the quantum of action 33 • Particles – a consequence of the quantum of action 34 • Quantum information 35 • Curiosities and fun chal- lenges about the quantum of action 36 • The dangers of buying a can of beans 37 • A summary: quantum physics, the law and indoctrination 39 40 2 Light – the strange consequences of the quantum of action How do faint lamps behave? 40 • Photons 44 • What is light? 46 • The size of copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net photons 47 • Are photons countable? – Squeezed light 47 • The positions of photons 51 • Are photons necessary? 54 • Interference: how can a wave be made up of particles? 56 • Interference of a single photon 59 • Reflection and dif- fraction deduced from photon arrows 60 • Refraction and partial reflection from photon arrows 62 • From photons to waves 63 • Can light move faster than light? – Real and virtual photons 64 • Indeterminacy of electric fields 65 • How can virtual photon exchange lead to attraction? 65 • Can two photons interfere? 66 • Curi- osities and fun challenges about photons 67 • A summary on light: particle and wave 69 72 3 Motion of matter – beyond classical physics Wine glasses, pencils and atoms – no rest 72 • No infinite measurement preci- sion 73 • Cool gas 73 • Flows and the quantization of matter 74 • Fluid flows and quantons 74 • Knocking tables and quantized conductivity 74 • Matter quan- tons and their motion – matter waves 76 • Mass and acceleration of quantons 79 • Why are atoms not flat? Why do shapes exist? 79 • Rotation, quantization of an- gular momentum, and the lack of north poles 81 • Rotation of quantons 82 • Silver, Stern and Gerlach – polarization of quantons 83 • Curiosities and fun chal- lenges about quantum matter 85 • First summary on the motion of quantum particles 86 87 4 The quantum description of matter and its motion States and measurements – the wave function 87 • Visualizing the wave function: rotating arrows and probability clouds 89 • The state evolution – the Schrödinger 12 contents equation 91 • Self-interference of quantons 93 • The speed of quantons 94 • Dis- persion of quantons 94 • Tunnelling and limits on memory – damping of quan- tons 95 • The quantum phase 97 • Can two electron beams interfere? Are there coherent electron beams? 101 • The least action principle in quantum physics 102 • The motion of quantons with spin 104 • Relativistic wave equations 105 • Bound motion, or composite vs. elementary quantons 107 • Curiosities and fun chal- lenges about quantum motion of matter 109 • A summary on motion of matter quantons 111 112 5 Permutation of particles – are particles like gloves? Distinguishing macroscopic objects 112 • Distinguishing atoms 113 • Why does indistinguishability appear in nature? 115 • Can quantum particles be coun- ted? 115 • What is permutation symmetry? 116 • Indistinguishability and wave function symmetry 117 • The behaviour of photons 118 • Bunching and anti- bunching 120 • The energy dependence of permutation symmetry 120 • Indis- tinguishability in quantum field theory 121 • How accurately is permutation sym- metry verified? 122 • Copies, clones and gloves 122 • Summary 124 Motion Mountain – The Adventure of Physics 125 6 Rotations and statistics – visualizing spin Quantum particles and symmetry 125 • Types of quantum particles 127 • Spin 1/2 and tethered objects 130 • The extension of the belt trick 133 • Angels, Pauli’s exclusion principle and the hardness of matter 135 • Is spin a rotation about an axis? 137 • Rotation requires antiparticles 138 • Why is fencing with laser beams impossible? 139 • Spin, statistics and composition 140 • The size and density of matter 141 • A summary on spin and indistinguishability 141 • Limits and open questions of quantum statistics 142 143 7 Superpositions and probabilities – quantum theory without copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ideology Why are people either dead or alive? 144 • Macroscopic superpositions, coherence and incoherence 144 • Decoherence is due to baths 146 • How baths lead to de- coherence – scattering 146 • How baths lead to decoherence – relaxation 148 • Summary on decoherence, life and death 150 • What is a system? What is an ob- ject? 151 • Entanglement 151 • Is quantum theory non-local? A bit about the Einstein–Podolsky–Rosen paradox 152 • Curiosities and fun challenges about superpositions 155 • Why do probabilities and wave function collapse appear in measurements? 157 • Why is ℏ necessary for probabilities? 162 • Hidden vari- ables 163 • Summary on probabilities and determinism 165 • What is the differ- ence between space and time? 167 • Are we good observers? 168 • What relates information theory, cryptology and quantum theory? 168 • Is the universe a com- puter? 169 • Does the universe have a wave function? And initial conditions? 169 171 8 Colours and other interactions between light and matter The causes of colour 171 • Using the rainbow to determine what stars are made of 180 • What determines the colours of atoms? 181 • The shape of atoms 185 • The size of atoms 186 • Relativistic hydrogen 188 • Relativistic wave equations – again 189 • Getting a first feeling for the Dirac equation 191 • Antimat- ter 192 • Virtual particles 193 • Curiosities and fun challenges about colour and atoms 194 • Material properties 196 • A tough challenge: the strength of electro- magnetism 196 • A summary on colours and materials 197 198 9 Quantum physics in a nutshell contents 13 Physical results of quantum theory 198 • Results on the motion of quantum particles 199 • Achievements in accuracy and precision 201 • Is quantum theory magic? 203 • Quantum theory is exact, but can do more 203 205 a Units, measurements and constants SI units 205 • The meaning of measurement 208 • Planck’s natural units 208 • Other unit systems 210 • Curiosities and fun challenges about units 211 • Preci- sion and accuracy of measurements 212 • Limits to precision 214 • Physical con- stants 214 • Useful numbers 221 223 b Numbers and vector spaces Numbers as mathematical structures 223 • Complex numbers 225 • Qua- ternions 227 • Octonions 233 • Other types of numbers 234 • From vector spaces to Hilbert spaces 235 • Mathematical curiosities and fun challenges 238 239 Challenge hints and solutions 247 Bibliography Motion Mountain – The Adventure of Physics 263 Credits Acknowledgements 263 • Film credits 264 • Image credits 264 267 Name index 274 Subject index copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net The Q uantum of Change In our quest to understand how things move, we discover that there is a smallest change value in nature, implying that motion is fuzzy, that boxes are never tight, that matter is composed of elementary units, and that light and interactions are streams of particles. The smallest change value explains why antimatter exists, why particles are unlike gloves, why copying machines do not exist, why probabilities are reasonable, and how all colours in nature are formed. Chapter 1 M I N I M UM AC T ION – QUA N T UM T H E ORY F OR P OET S “ Natura [in operationibus suis] non facit ” saltus.** 15th century C limbing Motion Mountain up to this point, we completed three legs. We Motion Mountain – The Adventure of Physics ame across Galileo’s mechanics (the description of motion for kids), then ontinued with Einstein’s relativity (the description of motion for science-fiction enthusiasts), and finally explored Maxwell’s electrodynamics (the description of mo- tion for business people). These three classical descriptions of motion are impressive, beautiful and useful. However, they have a small problem: they are wrong. The reason is simple: none of them describes life. Whenever we observe a flower or a butterfly, such as those of Figure 2, we enjoy the bright colours, the motion, the wild smell, the soft and delicate shape or the fine details of their symmetries. However, we know: copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ⊳ Classical physics cannot explain any characteristic length or time scale ob- served in nature. Now, flowers and animals – but also many non-living systems – have characteristic sizes, size ranges and proportions; and they have characteristic rhythms. And indeed, classical physics cannot explain their origin, because ⊳ The classical constants of nature – the gravitational constant 𝐺, the ideal gas constant 𝑅, the speed of light 𝑐, the vacuum permittivity 𝜀0 and the vacuum permeability 𝜇0 – do not allow defining length or time units: They cannot be combined to yield a length or time value. And they cannot be be used to build a meter bar. In fact, the classical constants do not even allow us to measure speed or force values, even though these measurements are fractions of 𝑐 and 𝑐4 /𝐺; because in order to meas- ure fractions, we need to define fractions first; however, defining fractions also requires length or time scales and units, which classical physics does not allow. Without measurements, there are also no emotions! Indeed, our emotions are triggered by our senses. And all the impressions and all the information that our senses Ref. 1 ** ‘Nature [in its workings] makes no jumps.’ 16 1 minimum action – quantum theory for poets Motion Mountain – The Adventure of Physics F I G U R E 2 Examples of quantum machines (© Linda de Volder). copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net provide us are – among others – measurements. Since classical physics does not provide measurement scales, we know: ⊳ Classical physics does not allow understanding senses or emotions. The reason for all these limitations is the following connection: ⊳ Classical physics alone cannot be used to build any measurement device. Every sense contains measurement devices. And every measurement device, like any pat- tern or rhythm, needs an internal scale, or, more generally, an internal measurement unit. Because classical physics does not provide any scale, classical physics does not explain how measurement devices work, not how senses work, and not how emotions appear. To understand emotions and life, we need to go beyond classical physics. Take any ex- ample of a pleasant situation,* such as a beautiful evening sky, a waterfall, a happy child Challenge 2 s or a caress. Classical physics is not able to explain any aspect of the situation: First, the colours and their origin remain mysterious. Secondly, all shapes, sizes and proportions remain mysterious. Thirdly, the timing and the duration of the involved processes can- * The photograph on page 14 shows a female glow worm, Lampyris noctiluca, as commonly found in the United Kingdom (© John Tyler, www.johntyler.co.uk/gwfacts.htm). 1 minimum action – quantum theory for poets 17 not be understood. Fourthly, all the sensations and emotions produced by the situation remain mysterious. To understand and explain these aspects, we need quantum theory. In fact, we will find out that both life and every type of pleasure are examples of quantum motion. Emotions are quantum processes. In the early days of physics, the impossibility to describe life and pleasure was not seen as a shortcoming, because neither senses nor material properties nor scales were thought to be related to motion. And pleasure was not considered a serious subject of investigation for a respectable researcher anyway. Today, the situation is different. In our Vol. I, page 411 adventure we have learned that our senses of time, hearing, touch, smell and sight are primarily detectors of motion. Without motion, there would be no senses. Furthermore, all detectors are made of matter. During the exploration on electromagnetism we began to understand that all properties of matter are due to motions of charged constituents. Density, stiffness, colour and all other material properties result from the electromag- Vol. III, page 231 netic behaviour of the Lego bricks of matter: namely, the molecules, the atoms and the electrons. Thus, the properties of matter are also consequences of motion. Moreover, Motion Mountain – The Adventure of Physics Vol. III, page 247 we saw that these tiny constituents are not correctly described by classical electrodyna- Vol. III, page 149 mics. We even found that light itself does not behave classically. Therefore the inability of classical physics to describe matter, light and the senses is indeed due to its intrinsic limitations. In fact, every failure of classical physics can be traced back to a single, fundamental Ref. 2 discovery made in 1899 by Max Planck:* ⊳ In nature, action values smaller than ℏ = 1.06 ⋅ 10−34 Js are not observed. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net All attempts to observe physical action values smaller than this fail.** In other words, in nature – as in a good cinema film – there is always some action. The existence of a smal- lest action value – the so-called quantum principle – is in complete contrast with clas- Challenge 3 s sical physics. (Why?) Despite this contrast, the quantum principle has passed an enorm- * Max Planck (1858–1947), professor of physics in Berlin, was a central figure in thermostatics and mod- ern physics. He discovered and named the Boltzmann constant 𝑘 and the quantum of action ℎ, often called Planck’s constant. His introduction of the quantum hypothesis gave birth to quantum theory. He also made the works of Einstein known in the physical community, and later organized a job for him in Berlin. He received the Nobel Prize for physics in 1918. He was an important figure in the German scientific estab- lishment; he also was one of the very few who had the courage to tell Adolf Hitler face to face that it was a bad idea to fire Jewish professors. (He got an outburst of anger as answer.) Famously modest, with many tragedies in his personal life, he was esteemed by everybody who knew him. ** In fact, this story is a slight simplification: the constant originally introduced by Planck was the (unre- duced) constant ℎ = 2πℏ. The factor 2π leading to the final quantum principle was added somewhat later, by other researchers. This somewhat unconventional, but didactically useful, approach to quantum theory is due to Niels Bohr. Ref. 3, Ref. 4 Nowadays, it is hardly ever encountered in the literature, despite its simplicity. Niels Bohr (b. 1885 Copenhagen, d. 1962 Copenhagen) was one of the great figures of modern physics. A daring thinker and a polite man, he made Copenhagen University into the new centre of development of quantum theory, overshadowing Göttingen. He developed the description of the atom in terms of quantum theory, for which he received the 1922 Nobel Prize in Physics. He had to flee Denmark in 1943 after the German invasion, because of his Jewish background, but returned there after the war, continuing to attract the best physicists across the world. 18 1 minimum action – quantum theory for poets F I G U R E 3 Max Planck (1858–1947) F I G U R E 4 Niels Bohr (1885–1962) Motion Mountain – The Adventure of Physics ous number of experimental tests, many of which we will encounter in this part of our mountain ascent. Above all, the quantum principle has never failed even a single test. The fundamental constant ℏ, which is pronounced ‘aitch-bar’, is called the quantum of action, or alternatively Planck’s constant. Planck discovered the quantum principle when Vol. III, page 149 studying the properties of incandescent light, i.e., of light emanating from hot bodies. But the quantum principle also applies to motion of matter, and even, as we will see later, to motion of empty space, such as gravitational waves. The quantum principle states that no experiment can measure an action smaller than ℏ. For a long time, Einstein tried to devise experiments to overcome this limit. But he copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net failed in all his attempts: nature does not allow it, as Bohr showed again and again. The same occurred to many other researchers. We recall that in physics – as in the theatre – action is a measure for the change oc- Vol. I, page 248 curring in a system. The quantum principle can thus rephrased as ⊳ In nature, a change smaller than ℏ = 1.06 ⋅ 10−34 Js cannot be observed. Therefore, a smallest action implies that there is a smallest change value in nature. If we compare two observations, there will always be change between them. Thus the quantum of action would perhaps be better named the quantum of change. Can a minimum change really exist in nature? To accept the idea, we need to explore three points, detailed in Table 1. We need to show that a smaller change is never observed in nature, show that smaller change values can never be observed, and finally, show that all consequences of this smallest change, however weird they may be, apply to nature. In fact, this exploration constitutes all of quantum physics. Therefore, these checks are all we do in the remaining of this part of our adventure. But before we explore some of the experiments that confirm the existence of a smallest change, we directly present some of its more surprising consequences. 1 minimum action – quantum theory for poets 19 TA B L E 1 How to convince yourself and others that there is a smallest action, or smallest change ℏ in nature. Compare this table with the two tables in volume II, that about maximum speed on page 26, and that about maximum force on page 109. S tat e m e n t Te s t The smallest action value ℏ is Check all observations. observer-invariant. Local change or action values < ℏ Check all observations. are not observed. Local change or action values < ℏ Check all attempts. cannot be produced. Local change or action values < ℏ Solve all paradoxes. cannot even be imagined. The smallest local change or action Deduce quantum theory Motion Mountain – The Adventure of Physics value ℏ is a principle of nature. from it. Show that all consequences, however weird, are confirmed by observation. The effects of the quantum of action on rest Since action is a measure of change, a minimum observable action means that two suc- cessive observations of the same system always differ by at least ℏ. In every system, there copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net is always something happening. As a consequence we find: ⊳ In nature there is no rest. Page 15 Everything moves, all the time, at least a little bit. Natura facit saltus.* True, these jumps are tiny, as ℏ is too small to be observable by any of our senses. Nevertheless, rest can be observed only macroscopically, and only as a long-time or many-particle average. For example, the quantum of action implies that in a mountain – an archetypal ‘system at rest’ – all the atoms and electrons are continually buzzing around. In short, ⊳ There is motion inside matter. Since there is a minimum action for all observers, and since there is no rest, we de- duce: ⊳ In nature there is no perfectly straight or perfectly uniform motion. Forget all you have learnt so far: Inertial motion is an approximation! An object can move in straight, uniform motion only approximately, and only when observed over long distances or long times. We will see later that the more massive the object is, the better * ‘Nature makes jumps.’ 20 1 minimum action – quantum theory for poets Challenge 4 s the approximation is. (Can you confirm this?) So macroscopic observers can still talk about space-time symmetries; and special relativity can thus be reconciled with quantum theory. Also free fall, or motion along a geodesic, exists only as a long-time average. So general relativity, which is based on the existence of freely-falling observers, cannot be correct when actions of the order of ℏ are involved. Indeed, the reconciliation of the quantum principle with general relativity – and thus with curved space – is a big chal- lenge. (The solution is simple only for weak, everyday fields.) The issues involved are so mind-shattering that they form a separate, final, part of this adventure. We thus explore situations without gravity first. The consequences of the quantum of action for objects Have you ever wondered why leaves are green? You probably know that they are green because they absorb blue (short-wavelength) and red (long-wavelength) light, while al- lowing green (medium-wavelength) light to be reflected. How can a system filter out the Motion Mountain – The Adventure of Physics small and the large, and let the middle pass through? To do so, leaves must somehow measure the frequency. But we have seen that classical physics does not allow measure- ment of time (or length) intervals, as any measurement requires a measurement unit, Vol. I, page 439 and classical physics does not allow such units to be defined. On the other hand, it takes only a few lines to confirm that with the help of the quantum of action ℏ (and the Boltzmann constant 𝑘, both of which Planck discovered), fundamental units for all measurable quantities can be defined, including time and therefore frequency. (Can you find a combination of the speed of light 𝑐, the gravitational constant 𝐺 and the quantum Challenge 5 s of action ℏ that gives a time? It will only take a few minutes.) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net In short, measurements are only possible at all because of the existence of the quantum of action. ⊳ All measurements are quantum effects. When Planck saw that the quantum of action allowed defining all units in nature, he was as happy as a child; he knew straight away that he had made a fundamental discovery, even though (in 1899) quantum theory did not yet exist. He even told his seven-year-old Ref. 5 son Erwin about it, while walking with him through the woods around Berlin. Planck explained to his son that he had made a discovery as important as universal gravity. Indeed, Planck knew that he had found the key to understanding many of the effects that were then unexplained. ⊳ In nature, all times and all frequencies are due to the quantum of action. All processes that take time are quantum processes. If you prefer, waiting is a quantum effect! In particular, without the quantum of action, oscillations and waves could not exist: ⊳ Every colour is a quantum effect. 1 minimum action – quantum theory for poets 21 But this is not all. Planck also realized that the quantum of action allows us to understand the size of all things. ⊳ Every size is a quantum effect. Challenge 6 e Can you find the combination of 𝑐, 𝐺 and ℏ that yields a length? With the quantum of action, it was finally possible to determine the maximum size of mountains, of trees and Vol. I, page 338 of humans. Planck knew that the quantum of action confirmed what Galileo had already deduced long before him: that sizes are due to fundamental, smallest scales in nature. Max Planck also understood that the quantum of action ℏ was the last missing con- stant of nature. With ℏ, it becomes possible to define a natural unit for every observable property in nature. Together, 𝑐, 𝐺 and ℏ allow to define units that are independent of culture or civilization – even extraterrestrials would understand them.* In short, ℏ al- lows understanding all observables. Therefore, with ℏ it is possible to draw the diagram shown in Figure 1 that encompasses all motion in nature, and thus all of physics. Motion Mountain – The Adventure of Physics In our environment, the size of all objects is related and due to the size of atoms. In turn, the size of atoms is a direct consequence of the quantum of action. Can you derive an approximation for the size of atoms, knowing that it is given by the motion of electrons Challenge 8 s of mass 𝑚e and charge 𝑒, constrained by the quantum of action? This connection, a simple formula, was discovered in 1910 by Arthur Erich Haas, 15 years before quantum theory was formulated. ⊳ Atom sizes are quantum effects. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net At the time, Haas was widely ridiculed.** Nowadays, his formula for the size of atoms is Page 186 found in all textbooks, including this one. In determining the size of atoms, the quantum of action has another important consequence: ⊳ Gulliver’s travels are impossible. There are no tiny people and no giant ones. Classically, nothing speaks against the idea; Challenge 9 s but the quantum of action prevents it. Can you supply the detailed argument? But if rest does not exist, how can shapes exist? Any shape of everyday life, including that of a flower, is the result of body parts remaining at rest with respect to each other. Now, all shapes result from interactions between the constituents of matter, as shown most clearly in the shapes of molecules. But how can a molecule, such as the water mo- lecule H2 O, shown in Figure 5, have a shape? In fact, a molecule does not have a fixed shape, but its shape fluctuates, as would be expected from the quantum of action. Des- pite the fluctuations, every molecule does have an average shape, because different angles and distances correspond to different energies. Again, these average length and angle val- * In fact, it is also possible to define all measurement units in terms of the speed of light 𝑐, the gravitational Challenge 7 s constant 𝐺 and the electron charge 𝑒. Why is this not fully satisfactory? ** Before the discovery of ℏ, the only simple length scale for the electron was the combination 𝑒2 /(4π𝜀0 𝑚e 𝑐2 ) ≈ 3 fm; this is ten thousand times smaller than an atom. We stress that any length scale containing 𝑒 is a quantum effect, and not a classical length scale, because 𝑒 is the quantum of electric charge. 22 1 minimum action – quantum theory for poets O H H F I G U R E 5 An artist’s impression of a water molecule made of two hydrogen (H) and one oxygen (O) atom. Motion Mountain – The Adventure of Physics F I G U R E 6 Max Born (1882 –1970) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ues only exist because the quantum of action yields fundamental length scales in nature. Without the quantum of action, there would be no shapes in nature. ⊳ All shapes are quantum effects. All shapes in everyday life are due to molecular shapes, or to their generalizations. The mass of an object is also a consequence of the quantum of action, as we will see later on. Since all material properties – such as density, colour, stiffness or polarizability – are defined as combinations of length, time and mass units, we find: ⊳ All material properties arise from the quantum of action. In short, the quantum of action determines the size, shape, colour, mass, and all other properties of objects, from stones to whipped cream. Why ‘ quantum ’ ? Quantum effects surround us on all sides. However, since the quantum of action is so small, its effects on motion appear mostly, but not exclusively, in microscopic systems. The study of such systems was called quantum mechanics by Max Born, one of the major 1 minimum action – quantum theory for poets 23 TA B L E 2 Some small systems in motion and the observed action values for their changes. System and change Action Motion Light Smallest amount of light absorbed by a coloured surface 1ℏ quantum Smallest impact when light reflects from mirror 2ℏ quantum Smallest consciously visible amount of light c. 5 ℏ quantum Smallest amount of light absorbed in flower petal 1ℏ quantum Blackening of photographic film c. 3 ℏ quantum Photographic flash c. 1017 ℏ classical Electricity Electron ejected from atom or molecule c. 1–2 ℏ quantum Electron extracted from metal c. 1–2 ℏ quantum Electron motion inside microprocessor c. 2–6 ℏ quantum Motion Mountain – The Adventure of Physics Signal transport in nerves, from one molecule to the next c. 5 ℏ quantum Current flow in lightning bolt c. 1038 ℏ classical Materials Tearing apart two neighbouring iron atoms c. 1–2 ℏ quantum Breaking a steel bar c. 1035 ℏ classical Basic process in superconductivity 1ℏ quantum Basic process in transistors 1ℏ quantum Basic magnetization process 1ℏ quantum Chemistry copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Atom collision in liquid at room temperature 1ℏ quantum Shape oscillation of water molecule c. 1 − 5 ℏ quantum Shape change of molecule, e.g. in chemical reaction c. 1 − 5 ℏ quantum Single chemical reaction curling a hair c. 2 − 6 ℏ quantum Tearing apart two mozzarella molecules c. 300 ℏ quantum Smelling one molecule c. 10 ℏ quantum Burning fuel in a cylinder in an average car engine explosion c. 1037 ℏ classical Life Air molecule hitting eardrum c. 2 ℏ quantum Smallest sound signal detectable by the ear Challenge 10 ny Single DNA duplication step during cell division c. 100 ℏ quantum Ovule fertilization c. 1014 ℏ classical Smallest step in molecular motor c. 5 ℏ quantum Sperm motion by one cell length c. 1015 ℏ classical Cell division c. 1019 ℏ classical Fruit fly’s wing beat c. 1024 ℏ classical Person walking one body length c. 2 ⋅ 1036 ℏ classical Nuclei and stars Nuclear fusion reaction in star c. 1 − 5 ℏ quantum Explosion of gamma-ray burster c. 1080 ℏ classical 24 1 minimum action – quantum theory for poets contributors to the field.* Later, the term quantum theory became more popular. Quantum theory arises from the existence of smallest measurable values in nature, generalizing the idea that Galileo had in the seventeenth century. As discussed in de- Vol. I, page 335 tail earlier on, it was Galileo’s insistence on ‘piccolissimi quanti’ – smallest quanta – of matter that got him into trouble. We will soon discover that the idea of a smallest change is necessary for a precise and accurate description of matter and of nature as a whole. Therefore Born adopted Galileo’s term for the new branch of physics and called it ‘Quantentheorie’ or ‘theory of quanta’. The English language adopted the Latin singular ‘quantum’ instead of the plural used in most other languages. Note that the term ‘quantum’ does not imply that all measurement values are multiples of a smallest one: this is so only in a few cases. Quantum theory is the description of microscopic motion. Quantum theory is neces- sary whenever a process produces an action value of the order of the quantum of action. Table 2 shows that all processes on atomic and molecular scales, including biological and chemical processes, are quantum processes. So are processes of light emission and Motion Mountain – The Adventure of Physics absorption. These phenomena can only be described with quantum theory. Table 2 also shows that the term ‘microscopic’ has a different meaning for a physicist and for a biologist. For a biologist, a system is ‘microscopic’ if it requires a microscope for its observation. For a physicist, a system is microscopic if its characteristic action is of the order of the quantum of action. In other words, for a physicist a system is usually mi- croscopic if it is not even visible in a (light) microscope. To increase the confusion, some quantum physicists nowadays call their own class of microscopic systems ‘mesoscopic’, while others call their systems ‘nanoscopic’. Both terms were introduced only to attract attention and funding: they are conceptually useless. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net The effect of the quantum of action on motion There is another way to characterize the difference between a microscopic, or quantum, system and a macroscopic, or classical, one. A smallest action implies that the difference between the action values 𝑆 of two successive observations of the same system, a time Δ𝑡 apart, cannot vanish. We have ℏ |𝑆(𝑡 + Δ𝑡) − 𝑆(𝑡)| = |(𝐸 ± Δ𝐸)(𝑡 + Δ𝑡) − 𝐸𝑡| = |𝐸Δ𝑡 ± 𝑡Δ𝐸 ± Δ𝐸Δ𝑡| ⩾ . (1) 2 * Max Born (b. 1882 Breslau, d. 1970 Göttingen) first studied mathematics, then turned to physics. A pro- fessor at Göttingen University, he made the city one of the world centres of physics. He developed quantum mechanics with his assistants Werner Heisenberg and Pascual Jordan, and then applied it to scattering, solid-state physics, optics and liquids. He was the first to understand that the wave function, or state func- Ref. 6 tion, describes a probability amplitude. Later, Born and Emil Wolf wrote what is still the main textbook on optics. Many of Born’s books were classics and read all over the world. Born attracted to Göttingen the most brilliant talents of the time, receiving as visitors Hund, Pauli, Nord- heim, Oppenheimer, Goeppert-Mayer, Condon, Pauling, Fock, Frenkel, Tamm, Dirac, Mott, Klein, Heitler, London, von Neumann, Teller, Wigner, and dozens of others. Being Jewish, Born lost his job in 1933, when criminals took over the German government. He emigrated, and became professor in Edinburgh, where he stayed for 20 years. Physics at Göttingen never recovered from this loss. For his elucidation of the meaning of the wave function he received the 1954 Nobel Prize in Physics. 1 minimum action – quantum theory for poets 25 F I G U R E 7 Werner Heisenberg (1901–1976) The factor 1/2 arises because a smallest action ℏ automatically implies an action inde- terminacy of half its value. Now the values of the energy 𝐸 and time 𝑡 – but not of (the Motion Mountain – The Adventure of Physics positive) Δ𝐸 or Δ𝑡 – can be set to zero if we choose a suitable observer. Thus, the ex- istence of a quantum of action implies that in any system the evolution is constrained by ℏ Δ𝐸Δ𝑡 ⩾ , (2) 2 where 𝐸 is the energy of the system and 𝑡 is its age, so that Δ𝐸 is the change of energy and Δ𝑡 is the time between two successive observations. Challenge 11 e By a similar reasoning, we find that for any physical system the position and mo- mentum are constrained by copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ℏ Δ𝑥Δ𝑝 ⩾ , (3) 2 where Δ𝑥 is the indeterminacy in position and Δ𝑝 is the indeterminacy in momentum. These two famous relations were called indeterminacy relations by their discoverer, Werner Heisenberg.* In English they are often called ‘uncertainty relations’; however, this term is incorrect. The quantities are not uncertain, but undetermined. Because of the quantum of action, system observables have no definite value. There is no way to ascribe * It is often said that the indeterminacy relation for energy and time has a different weight from that for momentum and position. This is a wrong idea, propagated by the older generation of physicists, which has survived through many textbooks for over 70 years. Just forget it. It is essential to remember that all four quantities appearing in the inequalities describe the internal properties of the system. In particular, 𝑡 is a time variable deduced from changes observed inside the system, and not the time coordinate measured by an outside clock; similarly, the position 𝑥 is not the external space coordinate, but the position characteriz- Ref. 7 ing the system. Werner Heisenberg (1901–1976) was an important theoretical physicist and an excellent table-tennis and tennis player. In 1925, as a young man, he developed, with some help from Max Born and Pascual Jordan, the first version of quantum theory; from it he deduced the indeterminacy relations. For these achievements he received the Nobel Prize in Physics in 1932. He also worked on nuclear physics and on turbulence. During the Second World War, he worked on the nuclear-fission programme. After the war, he published several successful books on philosophical questions in physics, slowly turned into a crank, and tried unsuccessfully – with some half-hearted help from Wolfgang Pauli – to find a unified description of nature based on quantum theory, the ‘world formula’. 26 1 minimum action – quantum theory for poets a precise value to momentum, position, or any other observable of a quantum system. We will use the term ‘indeterminacy relation’ throughout. The habit to call the relation a ‘principle’ is even more mistaken. Any system whose indeterminacy is of the order of ℏ is a quantum system; if the indeterminacy product is much larger, the system is classical, and then classical physics is sufficient for its description. So even though classical physics assumes that there are no measurement indeterminacies in nature, a system is classical only if its indeterminacies are large compared to the minimum possible ones! In other terms, quantum theory is necessary whenever we try to measure some quant- ity as precisely as possible. In fact, every measurement is itself a quantum process. And the indeterminacy relation implies that measurement precision is limited. The quantum of action shows that ⊳ Motion cannot be observed to infinite precision. Motion Mountain – The Adventure of Physics In other words, the microscopic world is fuzzy. This fact has many important con- sequences and many strange ones. For example, if motion cannot be observed with infin- ite precision, the very concept of motion needs to be handled with great care, as it cannot be applied in certain situations. In a sense, the rest of our quest is just an exploration of the implications of this result. In fact, as long as space-time is flat, it turns out that we can retain the concept of motion to describe observations, provided we remain aware of the limitations implied by the quantum principle. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net The surprises of the quantum of action The quantum of action ℏ implies a fuzziness of all motion. This fuzziness also implies the existence of short-time deviations from energy, momentum and angular-momentum conservation in microscopic systems. For general assurance it must be stressed that for long observation times – surely for all times longer than a microsecond – conservation Vol. I, page 238 holds. But in the first part of our adventure, we realized that any type of non-conservation implies the existence of surprises in nature. Well, here are some of them. Since precisely uniform motion does not exist, a system moving in one dimension only – such as the hand of a clock – always has the possibility of moving a bit in the opposite direction, thus leading to incorrect readings. Indeed, quantum theory predicts that clocks have essential limitations: ⊳ Perfect clocks do not exist. The deep implications of this statement will become clear step by step. It is also impossible to avoid that an object makes small displacement sideways. In fact, quantum theory implies that, strictly speaking, ⊳ Neither uniform nor one-dimensional motion exists. Also this statement harbours many additional surprises. 1 minimum action – quantum theory for poets 27 Quantum limitations apply also to metre rules. It is impossible to ensure that the rule is completely at rest with respect to the object being measured. Thus the quantum of action implies again, on the one hand, that measurements are possible, and on the other hand: ⊳ Measurement accuracy is limited. It also follows from the quantum of action that any inertial or freely-falling observer must be large, as only large systems approximate inertial motion. ⊳ An observer cannot be microscopic. If humans were not macroscopic, they could neither observe nor study motion. Because of the finite accuracy with which microscopic motion can be observed, we discover that Motion Mountain – The Adventure of Physics ⊳ Faster-than-light motion is possible in the microscopic domain. Quantum theory thus predicts tachyons, at least over short time intervals. For the same reason, ⊳ Motion backwards in time is possible over microscopic times and distances. In short, a quantum of action implies the existence of microscopic time travel. However, copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net this remains impossible in the macroscopic domain, such as everyday life. But there is more. Imagine a moving car suddenly disappearing for good. In such a situation, neither momentum nor energy would be conserved. The action change for such a disappearance is large compared to ℏ, so that its observation would contradict Challenge 12 s even classical physics – as you may wish to check. However, the quantum of action al- lows a microscopic particle, such as an electron, to disappear for a short time, provided it reappears afterwards. ⊳ The quantum of action implies that there is no permanence in nature. The quantum of action also implies: ⊳ The vacuum is not empty. If we look at empty space twice, the two observations being separated by a tiny time in- terval, some energy will be observed the second time. If the time interval is short enough, the quantum of action will lead to the observation of radiation or matter particles. In- deed, particles can appear anywhere from nowhere, and disappear just afterwards: the action limit requires it. In summary, nature exhibits short-term appearance and disap- pearance of matter and radiation. In other words, the classical idea of an empty vacuum is correct only when the vacuum is observed over a long time. The quantum of action implies that compass needles cannot work. If we look twice in 28 1 minimum action – quantum theory for poets E m p 0 Δx F I G U R E 8 Hills are never high enough. quick succession at a compass needle, or even at a house, we usually observe that it stays oriented in the same direction. But since physical action has the same dimensions as Motion Mountain – The Adventure of Physics Challenge 13 e angular momentum, a minimum value for action implies a minimum value for angular momentum. Even a macroscopic object has a minimum value for its rotation. In other words, quantum theory predicts ⊳ Everything rotates. An object can be non-rotating only approximately, when observations are separated by long time intervals. For microscopic systems, the quantum limits on rotation have specific effects. If the copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net rotation angle can be observed – as for molecules – the system behaves like a macroscopic object: its position and orientation are fuzzy. But for a system whose rotation angle can- not be observed, the quantum of action limits the angular momentum to multiples of ℏ/2. In particular, all microscopic bound systems – such as molecules, atoms, or nuclei – contain rotational motion and rotating components. Transformation, life and Demo critus At the beginning of our adventure, we mentioned that the Greeks distinguished three Vol. I, page 20 types of changes: transport, growth, and transformation. We also mentioned that Demo- critus had deduced that all these types of changes – including life and death – were in fact the same, and due to the motion of atoms. The quantum of action makes exactly this point. First of all, a minimum action implies that cages in zoos are dangerous and banks are not safe. A cage is a feature that needs a lot of energy to overcome. Physically speaking, the wall of a cage is an energy hill, resembling the real hill shown in Figure 8. Imagine that a particle with momentum 𝑝 approaches one side of the hill, which is assumed to have width Δ𝑥. In everyday life – and thus in classical physics – the particle will never be observed on the other side of the hill if its kinetic energy 𝑝2 /2𝑚 is less than the height 𝐸 of the hill. But imagine that the missing momentum to overcome the hill, Δ𝑝 = √2𝑚𝐸 − 𝑝, satisfies Δ𝑥Δ𝑝 ⩽ ℏ/2. The particle will have the possibility to overcome the hill, despite 1 minimum action – quantum theory for poets 29 E1 m E2 F I G U R E 9 Leaving enclosures. its insufficient energy. The quantum of action thus implies that a hill of width ℏ/2 Δ𝑥 ⩽ (4) √2𝑚𝐸 − 𝑝 Motion Mountain – The Adventure of Physics is not an obstacle to a particle of mass 𝑚. But this is not all. Since the value of the particle momentum 𝑝 is itself undetermined, a particle can overcome the hill even if the hill is wider than the value (4) – although the broader it is, the lower the probability will be. So any particle can overcome any obstacle. This is called the tunnelling effect, for obvious reasons. Classically, tunnelling is impossible. In quantum theory, the feat is possible, be- Page 89 cause the wave function does not vanish at the location of the hill; sloppily speaking, the wave function is non-zero inside the hill. It thus will be also non-zero behind the hill. As a result, quantum systems can penetrate or ‘tunnel’ through hills. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net In short, the minimum-action principle implies that there are no tight boxes in nature. Thanks to the tunnelling effect, ⊳ Matter is not impenetrable. The penetrability of all matter is in contrast to everyday, classical observation. Can you Challenge 14 s explain why lion cages work despite the quantum of action? By the way, the quantum of action also implies that a particle with a kinetic energy greater than the energy height of a hill can be reflected by the hill. Also this effect is impossible in classical physics. The minimum-action principle also implies that bookshelves are dangerous. Why? Shelves are obstacles to motion. A book on a shelf is in the same situation as the mass in Figure 9: the mass is surrounded by energy hills hindering its escape to the outer, lower- energy world. But thanks to the tunnelling effect, escape is always possible. The same picture applies to a branch of a tree, a nail in a wall, or anything attached to anything else. Things can never be permanently fixed together. In particular, we will discover that every example of light emission – even radioactivity – results from this effect. In summary, the quantum of action thus implies that ⊳ Decay is part of nature. Note that decay often appears in everyday life, under a different name: breaking. In fact, 30 1 minimum action – quantum theory for poets m m F I G U R E 10 Identical objects with crossing paths. Ref. 8 all breakages require the quantum of action for their description. Obviously, the cause of breaking is often classical, but the mechanism of breaking is always quantum. Only Motion Mountain – The Adventure of Physics objects that obey quantum theory can break. In short, there are no stable excited systems in nature. For the same reason, by the way, no memory can be perfect. (Can you confirm Challenge 15 s this?) Taking a more general view, ageing and death also result from the quantum of action. Death, like ageing, is a composition of breaking processes. When dying, the mechanisms in a living being break. Breaking is a form of decay, and is due to tunnelling. Death is thus a quantum process. Classically, death does not exist. Might this be the reason why Challenge 16 s so many people believe in immortality or eternal youth? We will also discover that the quantum of action is the reason for the importance of copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net the action observable in classical physics. In fact, the existence of a smallest action is the reason for the least-action principle of classical physics. A minimum action also implies that matter cannot be continuous, but must be com- posed of smallest entities. Indeed, any flow of a truly continuous material would contra- Challenge 17 s dict the quantum principle. Can you give the precise argument? Of course, at this point in our adventure, the non-continuity of matter is no longer a surprise. But the quantum of action implies that even radiation cannot be continuous. As Albert Einstein was the first to state clearly, light is made of quantum particles. Even more generally, the quantum of action implies that in nature ⊳ All flows and all waves are made of microscopic particles. The term ‘microscopic’ (or ‘quantum’) is essential, as such particles do not behave like little stones. We have already encountered several differences, and we will encounter oth- ers shortly. For these reasons, there should be a special name for microscopic particles; but so far all proposals, of which quanton is the most popular, have failed to catch on. The quantum of action has several strange consequences for microscopic particles. Take two such particles with the same mass and composition. Imagine that their paths cross, and that at the crossing they approach each other very closely, as shown in Fig- ure 10. A minimum action implies that in such a situation, if the distance becomes small enough, the two particles can switch roles, without anybody being able to avoid, or no- tice, it. Thus, in a volume of gas it is impossible – thanks to the quantum of action – to 1 minimum action – quantum theory for poets 31 M m1 m2 m m3 F I G U R E 11 Transformation through reaction. follow particles moving around and to say which particle is which. Can you confirm this Challenge 18 s deduction, and specify the conditions, using the indeterminacy relations? In summary Motion Mountain – The Adventure of Physics ⊳ In nature it is impossible to distinguish between identical particles. Challenge 19 s Can you guess what happens in the case of light? But matter deserves still more attention. Imagine again two particles – even two dif- ferent ones – approaching each other very closely, as shown in Figure 11. We know that if the approach distance gets small, things get fuzzy. Now, the minimum-action principle makes it possible for something to happen in that small domain as long as resulting out- going products have the same total linear momentum, angular momentum and energy as copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net the incoming ones. Indeed, ruling out such processes would imply that arbitrarily small actions could be observed, thus eliminating nature’s fuzziness, as you may wish to check Challenge 20 e for yourself. In short, ⊳ The quantum of action allows transformation of matter. One also says that the quantum of action allows particle reactions. In fact, we will dis- cover that all kinds of reactions in nature, including breathing, digestion, and all other chemical and nuclear reactions, are due just to the existence of the quantum of action. One type of process that is especially dear to us is growth. The quantum of action implies that all growth happens in small steps. Indeed, ⊳ All growth processes in nature are quantum processes. Above all, as mentioned already, the quantum of action explains life. Only the quantum of action makes reproduction and heredity possible. Birth, sexuality and death are con- sequences of the quantum of action. So Democritus was both right and wrong. He was right in deducing fundamental constituents for matter and radiation. He was right in unifying all change in nature – from transport to transformation and growth – as motion of particles. But he was wrong in assuming that the small particles behave like stones. As we will show in the following, the smallest particles behave like quantons: they behave randomly, and they behave partly 32 1 minimum action – quantum theory for poets as waves and partly as particles. R and omness – a consequence of the quantum of action What happens if we try to measure a change smaller than the quantum of action? Nature has a simple answer: we get random results. If we build an experiment that tries to pro- duce a change or action of the size of a quarter of the quantum of action, the experiment will produce, for example, a change of one quantum of action in a quarter of the cases, and no change in three quarters of the cases,* thus giving an average of one quarter of ℏ. ⊳ Attempts to measure actions below ℏ lead to random results. If you want to condense quantum physics in one key statement, this is it. The quantum of action leads to randomness at microscopic level. This connection can be seen also in the following way. Because of the indeterminacy relations, it is im- possible to obtain definite values for both the momentum and the position of a particle. Motion Mountain – The Adventure of Physics Obviously, definite values are also impossible for the individual components of an ex- perimental set-up or an observer. Therefore, initial conditions – both for a system and for an experimental set-up – cannot be exactly duplicated. The quantum of action thus implies that whenever an experiment on a microscopic system is performed twice, the outcomes will (usually) be different. The outcomes could only be the same if both the system and the observer were in exactly the same configuration each time. However, be- cause of the second principle of thermodynamics, and because of the quantum of action, reproducing a configuration is impossible. Therefore, copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ⊳ Microscopic systems behave randomly. Obviously, there will be some average outcome; but in all cases, microscopic observations are probabilistic. Many find this conclusion of quantum theory the most difficult to swal- low. But fact is: the quantum of action implies that the behaviour of quantum systems is strikingly different from that of classical systems. The conclusion is unavoidable: ⊳ Nature behaves randomly. Can we observe randomness in everyday life? Yes. Every window proves that nature be- haves randomly on a microscopic scale. Everybody knows that we can use a train window either to look at the outside landscape or, by concentrating on the reflected image, to ob- serve some interesting person inside the carriage. In other words, observations like that of Figure 12 show that glass reflects some of the light particles and lets some others pass through. More precisely, glass reflects a random selection of light particles; yet the aver- age proportion is constant. In these properties, partial reflection is similar to the tunnel- ling effect. Indeed, the partial reflection of photons in glass is a result of the quantum of action. Again, the situation can be described by classical physics, but the precise amount of reflection cannot be explained without quantum theory. We retain: * In this context, ’no change’ means ’no change’ in the physical variable to be measured; generally speaking, there is always some change, but not necessarily in the variable being measured. 1 minimum action – quantum theory for poets 33 F I G U R E 12 A famous quantum effect: how do train windows manage to show two superimposed images? (Photo © Greta Mansour) Motion Mountain – The Adventure of Physics F I G U R E 13 A particle and a screen with two nearby slits. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ⊳ Quantons move randomly. Without the quantum of action, train journeys would be much more boring. Waves – a consequence of the quantum of action The quantum of action implies an important result about the paths of particles. If a particle travels from one point to another, there is no way to say which path it has taken in between. Indeed, in order to distinguish between two possible, but slightly differ- ent, paths, actions smaller than ℏ would have to be measured reliably. In particular, if a particle is sent through a screen with two sufficiently close slits, as illustrated in Fig- ure 13, it is impossible to say which slit the particle passed through. This impossibility is fundamental. We already know phenomena of motion for which it is not possible to say with preci- sion how something moves or which path is taken behind two slits: waves behave in this Vol. I, page 314 way. All waves are subject to the indeterminacy relations 1 1 Δ𝜔Δ𝑡 ⩾ and Δ𝑘Δ𝑥 ⩾ . (5) 2 2 34 1 minimum action – quantum theory for poets A wave is a type of motion described by a phase that changes over space and time. This turns out to hold for all motion. In particular, this holds for matter. We saw above that quantum systems are subject to ℏ ℏ Δ𝐸Δ𝑡 ⩾ and Δ𝑝Δ𝑥 ⩾ . (6) 2 2 We are thus led to ascribe a frequency and a wavelength to a quantum system: 2π 𝐸 = ℏ𝜔 and 𝑝 = ℏ𝑘 = ℏ . (7) 𝜆 The energy–frequency relation for light and the equivalent momentum–wavelength re- lation were deduced by Max Planck in 1899. In the years from 1905 onwards, Albert Ein- stein confirmed that the relations are valid for all examples of emission and absorption of light. In 1923 and 1924, Louis de Broglie* predicted that the relation should hold also Motion Mountain – The Adventure of Physics for all quantum matter particles. The experimental confirmation came a few years later. Page 76 (This is thus another example of a discovery that was made about 20 years too late.) In short, the quantum of action implies: ⊳ Matter particles behave like waves. In particular, the quantum of action implies the existence of interference for streams of matter. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Particles – a consequence of the quantum of action The quantum of action, the smallest change, implies that flows cannot be arbitrarily weak. Vol. I, page 354 This applies to all flows: in particular, it applies to rivers, solid matter flows, gas flows, light beams, energy flows, entropy flows, momentum flows, angular momentum flows, probability flows, signals of all kind, electrical charge flows, colour charge flows and weak charge flows. Water flows in rivers, like any other matter flow, cannot be arbitrary small: the quantum of action implies that there is a smallest matter flow in nature. Depending on the situation, the smallest matter flow is a molecule, an atom or a smaller particle. In- deed, the quantum of action is also at the origin of the observation of a smallest charge in electric current. Since all matter can flow, the quantum of action implies: ⊳ All matter has particle aspects. * Louis de Broglie (b. 1892 Dieppe, d. 1987 Paris), physicist and professor at the Sorbonne. The energy– frequency relation for light had earned Max Planck and Albert Einstein the Nobel Prize in Physics, in 1918 and 1921. De Broglie expanded the relation to predict the wave nature of the electron (and of all other quantum matter particles): this was the essence of his doctoral thesis. The prediction was first confirmed experimentally a few years later, in 1927. For the prediction of the wave nature of matter, de Broglie received the Nobel Prize in Physics in 1929. Being an aristocrat, he did no more research after that. For example, it was Schrödinger who then wrote down the wave equation, even though de Broglie could equally have done so. 1 minimum action – quantum theory for poets 35 In the same way, the quantum of action, the smallest change, implies that light cannot be arbitrarily faint. There is a smallest illumination in nature; it is called a photon or a light quantum. Now, light is a wave, and the argument can be made for any other wave as well. In short, the quantum of action thus implies: ⊳ All waves have particle aspects. This has been proved for light waves, water waves, X-rays, sound waves, plasma waves, fluid whirls and any other wave type that has ever been observed. There is one exception: gravitational waves have finally been observed in 2016, many decades after their predic- tion; it is expected that their particle-like aspects, the gravitons, also exist, though this might take a long time to prove by experiment. In summary, the quantum of action states: ⊳ If something moves, it is made of quantum particles, or quantons. Motion Mountain – The Adventure of Physics Later on we will explore and specify the exact differences between a quantum particle and a small stone or a grain of sand. We will discover that matter quantons move differently, behave differently under rotation, and behave differently under exchange. Q uantum information In computer science, the smallest unit of change is called a ‘bit change’. The existence of a smallest change in nature implies that computer science – or information science – can be used to describe nature, and in particular quantum theory. This analogy has attracted copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net much research in the past decades, and explored many interesting questions: Is unlimited information storage possible? Can information be read out and copied completely? Can information be transmitted while keeping it secret? Can information transmission and storage be performed independently of noise? Can quantum physics be used to make new types of computers? So far, the answer to all these questions is negative; but the hope to change the situation is not dead yet. The analogy between quantum theory and information science is limited: information science can describe only the ‘software’ side of devices. For a physicist, the ‘hardware’ side of nature is central. The hardware of nature enters the description whenever the actual value ℏ of the quantum of action must be introduced. As we explore the similarities and differences between nature and information sci- ence, we will discover that the quantum of action implies that macroscopic physical sys- tems cannot be copied – or ‘cloned’, as quantum theorists like to say. Nature does not allow copies of macroscopic objects. In other words: ⊳ Perfect copying machines do not exist. The quantum of action makes it impossible to gather and use all information in a way that allows production of a perfect copy. The exploration of copying machines will remind us again that the precise order in which measurements are performed in an experiment matters. When the order 36 1 minimum action – quantum theory for poets of measurements can be reversed without affecting the net result, physicists speak of ‘commutation’. The quantum of action implies: ⊳ Physical observables do not commute. We will also find that the quantum of action implies that systems are not always Page 152 independent, but can be entangled. This term, introduced by Erwin Schrödinger, de- scribes one of the most absurd consequences of quantum theory. Entanglement makes everything in nature connected to everything else. Entanglement produces effects that seem (but are not) faster than light. ⊳ Entanglement produces a (fake) form of non-locality. Entanglement implies that trustworthy communication cannot exist. Motion Mountain – The Adventure of Physics Ref. 9 We will also discover that decoherence is an ubiquitous process in nature that influ- ences all quantum systems. For example, it allows measurements on the one hand and Page 157 makes quantum computers impossible on the other. Curiosities and fun challenges ab ou t the quantum of action Even if we accept that no experiment performed so far contradicts the minimum action, we still have to check that the minimum action does not contradict reason. In particular, the minimum action must also be consistent with all imagined experiments. This is not copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net self-evident. ∗∗ Challenge 21 s Where is the quantum scale in a pendulum clock? ∗∗ When electromagnetic fields come into play, the value of the action (usually) depends on the choice of the vector potential, and thus on the choice of gauge. We saw in the part Vol. III, page 85 on electrodynamics that a suitable choice of gauge can change the value of the action by adding or subtracting any desired amount. Nevertheless, there is a smallest action in nature. This is possible, because in quantum theory, physical gauge changes cannot add or subtract any amount, but only multiples of twice the minimum value. Thus they do not allow us to go below the minimum action. ∗∗ Adult plants stop growing in the dark. Without light, the reactions necessary for growth Challenge 22 s cease. Can you show that this is a quantum effect, not explainable by classical physics? ∗∗ Most quantum processes in everyday life are electromagnetic. Can you show that the quantum of action must also hold for nuclear processes, i.e., for processes that are not Challenge 23 s electromagnetic? 1 minimum action – quantum theory for poets 37 ∗∗ Challenge 24 s Is the quantum of action independent of the observer, even near the speed of light? This question was the reason why Planck contacted the young Einstein, inviting him to Berlin, thus introducing him to the international physics community. ∗∗ The quantum of action implies that tiny people, such as Tom Thumb, cannot exist. The quantum of action implies that fractals cannot exist in nature. The quantum of action implies that ‘Moore’s law’ of semiconductor electronics, which states that the number of Challenge 25 s transistors on a chip doubles every two years, cannot be valid for ever. Why not? ∗∗ Take a horseshoe. The distance between the two ends is not fixed, since otherwise their position and velocity would be known at the same time, contradicting the indeterminacy relation. Of course, this reasoning is also valid for any other solid object. In short, both Motion Mountain – The Adventure of Physics quantum mechanics and special relativity show that rigid bodies do not exist, albeit for different reasons. ∗∗ Angular momentum has the same dimensions as action. A smallest action implies that there is a smallest angular momentum in nature. How can this be, given that some Challenge 26 s particles have spin zero, i.e., have no angular momentum? ∗∗ copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Could we have started the whole discussion of quantum theory by stating that there is a Challenge 27 s minimum angular momentum instead of a minimum action? ∗∗ Niels Bohr, besides propagating the idea of a minimum action, was also an enthusiast of the so-called complementarity principle. This is the idea that certain pairs of observables of a system – such as position and momentum – have linked precision: if one observable of the pair is known to high precision, the other observable is necessarily known with Challenge 28 s low precision. Can you deduce this principle from the minimum action? The dangers of buying a can of beans Another way to show the absurd consequences of quantum theory is given by the ul- timate product warning, which according to certain well-informed lawyers should be Ref. 10 printed on every can of beans and on every product package. It shows in detail how deeply our human condition fools us. Warning: care should be taken when looking at this product: It emits heat radiation. Bright light has the effect to compress this product. Warning: care should be taken when touching this product: 38 1 minimum action – quantum theory for poets Part of it could heat up while another part cools down, causing severe burns. Warning: care should be taken when handling this product: This product consists of at least 99.999 999 999 999 % empty space. This product contains particles moving with speeds higher than one million kilo- metres per hour. Every kilogram of this product contains the same amount of energy as liberated by about one hundred nuclear bombs.* In case this product is brought in contact with antimatter, a catastrophic explosion will occur. In case this product is rotated, it will emit gravitational radiation. Warning: care should be taken when transporting this product: The force needed depends on its velocity, as does its weight. This product will emit additional radiation when accelerated. Motion Mountain – The Adventure of Physics This product attracts, with a force that increases with decreasing distance, every other object around, including its purchaser’s kids. Warning: care should be taken when storing this product: It is impossible to keep this product in a specific place and at rest at the same time. Except when stored underground at a depth of several kilometres, over time cosmic radiation will render this product radioactive. This product may disintegrate in the next 1035 years. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net It could cool down and lift itself into the air. This product warps space and time in its vicinity, including the storage container. Even if stored in a closed container, this product is influenced and influences all other objects in the universe, including your parents in law. This product can disappear from its present location and reappear at any random place in the universe, including your neighbour’s garage. Warning: care should be taken when travelling away from this product: It will arrive at the expiration date before the purchaser does so. Warning: care should be taken when using this product: Any use whatsoever will increase the entropy of the universe. The constituents of this product are exactly the same as those of any other object in the universe, including those of rotten fish. All these statements are correct. The impression of a certain paranoid side to quantum physics is purely coincidental. * A standard nuclear warhead has an explosive yield of about 0.2 megatons (implied is the standard explosive Ref. 11 trinitrotoluene or TNT), about thirteen times the yield of the Hiroshima bomb, which was 15 kilotonne. A megatonne is defined as 1 Pcal=4.2 PJ, even though TNT delivers about 5 % slightly less energy than this value. In other words, a megaton is the energy content of about 47 g of matter. That is less than a handful for most solids or liquids. 1 minimum action – quantum theory for poets 39 A summary: quantum physics, the law and ind o ctrination The mere existence of a quantum of action, a quantum of change, has many deep con- sequences: randomness, wave-particle duality, matter transformation, death, and, above all, new thinking habits. Don’t all the deductions from the quantum of action presented so far look wrong, or at least crazy? In fact, if you or your lawyer made some of the statements on quantum physics in court, maybe even under oath, you might end up in prison! However, all the above statements are correct: they are all confirmed by experiment. And there are many more surprises to come. You may have noticed that, in the preceding examples, we have made no explicit reference to electricity, to the nuclear interactions or to gravity. In these domains the surprises are even more astonishing. Observation of antimatter, electric cur- rent without resistance, the motion inside muscles, vacuum energy, nuclear reactions in stars, and – maybe one day – the boiling of empty space, will fascinate you as much as they have fascinated, and still fascinate, thousands of researchers. In particular, the consequences of the quantum of action for the early universe are Motion Mountain – The Adventure of Physics Challenge 29 d mind-boggling. Just try to explore for yourself its consequences for the big bang. To- gether, all these topics will lead us a long way towards the aim of our adventure. The consequences of the quantum of action are so strange, so incredible, and so numerous, that quantum physics can rightly be called the description of motion for crazy scientists. In a sense, this generalizes our previous definition of quantum physics as the description of motion related to pleasure. Unfortunately, it is sometimes claimed that ‘nobody understands quantum theory’. Page 167 This is wrong. In fact, it is worse than wrong: it is indoctrination and disinformation. Indoctrination and disinformation are methods that prevent people from making up copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net their own mind and from enjoying life. In reality, the consequences of the quantum of action can be understood and enjoyed by everybody. In order to do so, our first task on our way towards completing our adventure will be to use the quantum of action to study our classical standard of motion: the motion of light. “ Nie und nirgends hat es Materie ohne ” Bewegung gegeben, oder kann es sie geben. Friedrich Engels, Anti-Dühring.* Ref. 12 * ‘Never and nowhere has matter existed, nor can it exist, without motion.’ Friedrich Engels (1820–1895) was one of the theoreticians of Marxism. Chapter 2 L IG H T – T H E ST R A NG E C ON SE QU E NC E S OF T H E QUA N T UM OF AC T ION “ Alle Wesen leben vom Lichte, ” jedes glückliche Geschöpfe. Friedrich Schiller, Wilhelm Tell.** S ince all the colours of materials are quantum effects, it becomes mandatory to Motion Mountain – The Adventure of Physics tudy the properties of light itself. If a smallest change really exists, then there hould also be a smallest illumination in nature. This conclusion was already drawn Ref. 13 in ancient Greece, for example by Epicurus (341–271 b ce), who stated that light is a stream of little particles. The smallest possible illumination would then be that due to a single light particle. Today, the particles are called light quanta or photons. Incredibly, Epicurus himself could have checked his prediction with an experiment. How d o faint lamps behave? Ref. 14 Around 1930, Brumberg and Vavilov found a beautiful way to check the existence of copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net photons using the naked eye and a lamp. Our eyes do not allow us to consciously detect single photons, but Brumberg and Vavilov found a way to circumvent this limitation. In fact, the experiment is so simple that it could have been performed many centuries earlier; but nobody had had a sufficiently daring imagination to try it. Brumberg and Vavilov constructed a mechanical shutter that could be opened for time intervals of 0.1 s. From the other side, in a completely dark room, they illuminated the opening with extremely weak green light: about 200 aW at 505 nm, as shown in Fig- ure 14. At that intensity, whenever the shutter opens, on average about 50 photons can pass. This is just the sensitivity threshold of the eye. To perform the experiment, they repeatedly looked into the open shutter. The result was simple but surprising. Some- times they observed light, and sometimes they did not. Whether they did or did not was completely random. Brumberg and Vavilov gave the simple explanation that at low lamp powers, because of fluctuations, the number of photons is above the eye threshold half the time, and below it the other half. The fluctuations are random, and so the conscious de- tection of light is as well. This would not happen if light were a continuous stream: in that case, the eye would detect light at each and every opening of the shutter. (At higher light intensities, the percentage of non-observations quickly decreases, in accordance with the explanation given.) In short, a simple experiment proves: ** ‘From light all beings live, each fair-created thing.’ Friedrich Schiller (b. 1759 Marbach, d. 1805 Weimar), poet, playwright and historian. 2 light – and the quantum of action 41 lamp strong shutter head, after filter 45 minutes in complete darkness F I G U R E 14 How to experience single photon effects (see text). Motion Mountain – The Adventure of Physics photographic glass film F I G U R E 15 How does a white-light spectrum appear at white red extremely long screen distances? green (The short-screen-distance violet spectrum shown, © Andrew Young, is optimized for CRT display, not for colour printing, as explained on mintaka.sdsu.edu/ copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net GF/explain/optics/rendering. html.) ⊳ Light is made of photons. Nobody knows how the theory of light would have developed if this simple experiment had been performed 100 or even 2500 years earlier. The reality of photons becomes more convincing if we use devices to help us. A simple way is to start with a screen behind a prism illuminated with white light, as shown in Figure 15. The light is split into colours. As the screen is placed further and further away, the illumination intensity cannot become arbitrarily small, as that would contradict the quantum of action. To check this prediction, we only need some black-and-white photo- graphic film. Film is blackened by daylight of any colour; it becomes dark grey at medium intensities and light grey at lower intensities. Looking at an extremely light grey film un- der the microscope, we discover that, even under uniform illumination, the grey shade is actually composed of black spots, arranged more or less densely. All these spots have the same size, as shown in Figure 16. This regular size suggests that a photographic film reacts to single photons. Detailed research confirms this conjecture; in the twentieth century, the producers of photographic films have elucidated the underlying atomic mechanism in all its details. 42 2 light – and the quantum of action F I G U R E 16 Exposed photographic ﬁlm at increasing magniﬁcation (© Rich Evans). Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net F I G U R E 17 Detectors that allow photon counting: photomultiplier tubes (left), an avalanche photodiode (top right, c. 1 cm) and a microchannel plate (bottom right, c. 10 cm) (© Hamamatsu Photonics). Single photons can be detected most elegantly with electronic devices. Such devices Ref. 15 can be photomultipliers, photodiodes, microchannel plates or rod cells in the eye; a se- lection is shown in Figure 17. Also these detectors show that low-intensity light does not produce a homogeneous colour: on the contrary, low-intensity produces a random pat- tern of equal spots, even when observing typical wave phenomena such as interference patterns, as shown in Figure 18. Today, recording and counting individual photons is a standard experimental procedure. Photon counters are part of many spectroscopy set- ups, such as those used to measure tiny concentrations of materials. For example, they are used to detect drugs in human hair. All experiments thus show the same result: whenever sensitive light detectors are con- structed with the aim of ‘seeing’ as accurately as possible – and thus in environments as 2 light – and the quantum of action 43 Motion Mountain – The Adventure of Physics F I G U R E 18 Light waves are made of particles: observation of photons – black spots in these negatives – in a low intensity double slit experiment, with exposure times of 1, 2 and 5 s, using an image intensiﬁer (© Delft University of Technology). light detectors copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net radiating atom F I G U R E 19 An atom radiating one photon triggers only one detector and recoils in only one direction. dark as possible – one finds that light manifests as a stream of light quanta. Nowadays they are usually called photons, a term that appeared in 1926. Light of low or high intens- ity corresponds to a stream with a small or large number of photons. A particularly interesting example of a low-intensity source of light is a single atom. Atoms are tiny spheres. When atoms radiate light or X-rays, the radiation should be emit- ted as a spherical wave. But in all experiments – see Figure 19 for a typical set-up – the light emitted by an atom is never found to form a spherical wave, in contrast to what we might expect from everyday physics. Whenever a radiating atom is surrounded by many detectors, only a single detector is triggered. Only the average over many emissions and detections yields a spherical shape. The experiments shows clearly that partial photons cannot be detected. All experiments in dim light thus show that the continuum description of light is 44 2 light – and the quantum of action incorrect. All such experiments thus prove directly that light is a stream of particles, as Epicurus had proposed in ancient Greece. More precise measurements confirm the role of the quantum of action: every photon leads to the same amount of change. All photons of the same frequency blacken a film or trigger a scintillation screen in the same way. In short, the amount of change induced by a single photon is indeed the smallest amount of change that light can produce. If there were no smallest action value, light could be packaged into arbitrarily small amounts. But nature is different. In simple terms: the classical description of light by a Vol. III, page 86 continuous vector potential 𝐴(𝑡, 𝑥), or electromagnetic field 𝐹(𝑡, 𝑥), whose evolution is described by a principle of least action, is wrong. Continuous functions do not describe the observed particle effects. A modified description is required. The modification has to be significant only at low light intensities, since at high, everyday intensities the classical Lagrangian describes all experimental observations with sufficient accuracy.* At which intensities does light cease to behave as a continuous wave? Human eyesight does not allow us to consciously distinguish single photons, although experiments show Motion Mountain – The Adventure of Physics Ref. 16 that the hardware of the eye is in principle able to do so. The faintest stars that can be seen at night produce a light intensity of about 0.6 nW/m2 . Since the pupil of the eye is small, and we are not able to see individual photons, photons must have energies smaller than 100 aJ. Brumberg and Vavilov’s experiment yields an upper limit of around 20 aJ. An exact value for the quantum of action found in light must be deduced from labor- atory experiment. Some examples are given in the following. Photons In general, all experiments show that a beam of light of frequency 𝑓 or angular frequency copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 𝜔, which determines its colour, is accurately described as a stream of photons, each with the same energy 𝐸 given by 𝐸 = ℏ 2π𝑓 = ℏ 𝜔 . (8) This relation was first deduced by Max Planck in 1899. He found that for light, the smal- lest measurable action is given by the quantum of action ℏ. In short, colour is a property of photons. A coloured light beam is a hailstorm of corresponding photons. Vol. III, page 149 The value of Planck’s constant can be determined from measurements of black bodies Page 214 or other light sources. All such measurements coincide and yield ℏ = 1.054 571 726(47) ⋅ 10−34 Js , (9) a value so small that we can understand why photons go unnoticed by humans. For ex- Challenge 30 e ample, a green photon with a wavelength of 555 nm has an energy of 0.37 aJ. Indeed, in normal light conditions the photons are so numerous that the continuum approximation for the electromagnetic field is highly accurate. In the dark, the insensitivity of the signal processing of the human eye – in particular the slowness of the light receptors – makes Ref. 16 photon counting impossible. However, the eye is not far from the maximum possible Challenge 31 ny sensitivity. From the numbers given above about dim stars, we can estimate that humans * The transition from the classical case to the quantum case used to be called quantization. This concept, and the ideas behind it, are only of historical interest today. 2 light – and the quantum of action 45 are able to see consciously, under ideal conditions, flashes of about half a dozen photons; in normal conditions, the numbers are about ten times higher. Let us explore the other properties of photons. Above all, photons have no measurable Challenge 32 s (rest) mass and no measurable electric charge. Can you confirm this? In fact, experiments can only provide an upper limit for both quantities. The present experimental upper limit Ref. 17 for the (rest) mass of a photon is 10−52 kg, and for the charge is 5⋅10−30 times the electron charge. These limits are so small that we can safely say that both the mass and the charge of the photon vanish. We know that intense light can push objects. Since the energy, the lack of mass and Challenge 33 e the speed of photons are known, we deduce that the photon momentum is given by 𝐸 2π 𝑝= =ℏ or 𝑝 = ℏ 𝑘 . (10) 𝑐 𝜆 In other words, if light is made of particles, we should be able to play billiard with them. Motion Mountain – The Adventure of Physics Ref. 18 This is indeed possible, as Arthur Compton showed in a famous experiment in 1923. He directed X-rays, which are high-energy photons, onto graphite, a material in which electrons move almost freely. He found that whenever the electrons in the material are hit by the X-ray photons, the deflected X-rays change colour. His experiment is shown in Figure 20. As expected, the strength of the hit is related to the deflection angle of the photon. From the colour change and the deflection angle, Compton confirmed that the photon momentum indeed satisfies the expression 𝑝 = ℏ 𝑘. All other experiments agree that photons have momentum. For example, when an atom emits light, the atom feels a recoil. The momentum again turns out to be given by the expression 𝑝 = ℏ 𝑘. In short, the quantum of action determines the momentum of copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net the photon. The value of a photon’s momentum respects the indeterminacy relation. Just as it is impossible to measure exactly both the wavelength of a wave and the position of its crest, so it is impossible to measure both the momentum and the position of a photon. Can you Challenge 34 s confirm this? In other words, the value of the photon momentum is a direct consequence of the quantum of action. From our study of classical physics, we know that light has a property beyond its col- our: light can be polarized. That is only a complicated way to say that light can turn Vol. III, page 123 the objects that it shines on. In other words, light has an angular momentum oriented (mainly) along the axis of propagation. What about photons? Measurements consistently find that each light quantum carries an angular momentum given by 𝐿 = ℏ. It is called its helicity. The quantity is similar to one found for massive particles: one therefore also speaks of the spin of a photon. In short, photons somehow ‘turn’ – in a direction either parallel or antiparallel to their direction of motion. Again, the magnitude of the photon helicity, or spin, is no surprise; it confirms the classical relation 𝐿 = 𝐸/𝜔 between energy Vol. III, page 123 and angular momentum that we found in the section on classical electrodynamics. Note that, counterintuitively, the angular momentum of a single photon is fixed, and thus in- dependent of its energy. Even the most energetic photons have 𝐿 = ℏ. Of course, the value of the helicity also respects the limit given by the quantum of action. The many consequences of the helicity (spin) value ℏ will become clear in the following. 46 2 light – and the quantum of action X-ray detector deflected photon with photon after wavelength λ deflection the collision, angle with wave- length λ+Δλ X-ray collision source in X-ray sample X-ray electron sample source detector after the collision F I G U R E 20 A modern version of Compton’s experiment ﬁts on a table. The experiment shows that photons have momentum: X-rays – and thus the photons they consist of – change frequency when they hit the electrons in matter in exactly the same way as predicted from colliding particles (© Helene Hoffmann). Motion Mountain – The Adventure of Physics What is light? “ La lumière est un mouvement luminaire de ” corps lumineux. Blaise Pascal* In the seventeenth century, Blaise Pascal used the above statement about light to make fun of certain physicists, ridiculing the blatant use of a circular definition. Of course, he was right: in his time, the definition was indeed circular, as no meaning could be given to any of the terms. But whenever physicists study an observation with care, philosophers copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net lose out. All those originally undefined terms now have a definite meaning and the cir- cular definition is resolved. Light is indeed a type of motion; this motion can rightly be called ‘luminary’ because, in contrast to the motion of material bodies, it has the unique property 𝑣 = 𝑐; the luminous bodies, called light quanta or photons, are characterized, and differentiated from all other particles, by their dispersion relation 𝐸 = 𝑐𝑝, their en- ergy 𝐸 = ℏ𝜔, their spin 𝐿 = ℏ, the vanishing of all other quantum numbers, and the property of being the quanta of the electromagnetic field. In short, light is a stream of photons. It is indeed a ‘luminary movement of luminous bodies’. Photons provide our first example of a general property of the world on small scales: all waves and all flows in nature are made of quantum particles. Large numbers of (coherent) quantum particles – or quantons – behave as and form waves. We will see shortly that this is the case even for matter. Quantons are the fundamental constituents of all waves and all flows, without exception. Thus, the everyday continuum description of light is similar in many respects to the description of water as a continuous fluid: photons are the atoms of light, and continuity is an approximation valid for large numbers of particles. Single quantons often behave like classical particles. Physics books used to discuss at length a so-called wave–particle duality. Let us be clear from the start: quantons, or quantum particles, are neither classical waves nor clas- * ‘Light is the luminary movement of luminous bodies.’ Blaise Pascal (b. 1623 Clermont, d. 1662 Paris), important mathematician and physicist up to the age of 26, after which he became a theologian and philo- sopher. 2 light – and the quantum of action 47 sical particles. In the microscopic world, quantons are the fundamental objects. However, there is much that is still unclear. Where, inside matter, do these mono- chromatic photons come from? Even more interestingly, if light is made of quantons, all electromagnetic fields, even static ones, must be made of photons as well. However, in static fields nothing is flowing. How is this apparent contradiction resolved? And what implications does the particle aspect have for these static fields? What is the difference between quantons and classical particles? The properties of photons require more careful study. The size of photons First of all, we might ask: what are these photons made of? All experiments so far, per- formed down to the present limit of about 10−20 m, give the same answer: ‘we can’t find anything’. This is consistent with both a vanishing mass and a vanishing size of photons. Indeed, we would intuitively expect a body with a finite size to have a finite mass. Thus, although experiments can give only an upper limit, it is consistent to claim that a photon Motion Mountain – The Adventure of Physics has zero size. A particle with zero size cannot have any constituents. Thus a photon cannot be di- vided into smaller entities: photons are not composite. For this reason, they are called elementary particles. We will soon give some further strong arguments for this result. Challenge 35 s (Can you find one?) Nevertheless, the conclusion is strange. How can a photon have vanishing size, have no constituents, and still be something? This is a hard question; the answer will appear only in the last volume of our adventure. At the moment we simply have to accept the situation as it is. We therefore turn to an easier question. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Are photons countable? – S queezed light “ ” Also gibt es sie doch. Max Planck* We saw above that the simplest way to count photons is to distribute them across a large screen and then to absorb them. But this method is not entirely satisfactory, as it destroys the photons. How can we count photons without destroying them? One way is to reflect photons in a mirror and measure the recoil of the mirror. It seems almost unbelievable, but nowadays this effect is becoming measurable even for small numbers of photons. For example, it has to be taken into account in relation to the Vol. II, page 181 laser mirrors used in gravitational wave detectors, whose position has to be measured with high precision. Another way of counting photons without destroying them involves the use of special high-quality laser cavities. It is possible to count photons by the effect they have on atoms cleverly placed inside such a cavity. In other words, light intensity can indeed be measured without absorption. These measurement show an important issue: even the best light beams, from the most sophist- * ‘Thus they do exist after all.’ Max Planck, in his later years, said this after standing silently, for a long time, in front of an apparatus that counted single photons by producing a click for each photon it detected. For a large part of his life, Planck was sceptical of the photon concept, even though his own experiments and conclusions were the starting point for its introduction. 48 2 light – and the quantum of action icated lasers, fluctuate in intensity. There are no steady beams. This comes as no surprise: if a light beam did not fluctuate, observing it twice would yield a vanishing value for the action. However, there is a minimum action in nature, namely ℏ. Thus any beam and any flow in nature must fluctuate. But there is more. A light beam is described, in a cross section, by its intensity and phase. The change – or action – that occurs while a beam propagates is given by the product of intensity and phase. Experiments confirm the obvious deduction: the intensity and phase of a beam behave like the momentum and position of a particle in that they obey an indeterminacy relation. You can deduce it yourself, in the same way as we deduced Heisenberg’s rela- tions. Using as characteristic intensity 𝐼 = 𝐸/𝜔, the beam energy divided by the angular frequency, and calling the phase 𝜑, we get* ℏ Δ𝐼 Δ𝜑 ⩾ . (12) 2 Motion Mountain – The Adventure of Physics Equivalently, the indeterminacy product for the average photon number 𝑛 = 𝐼/ℏ = 𝐸/ℏ𝜔 and the phase 𝜑 obeys: 1 Δ𝑛 Δ𝜑 ⩾ . (13) 2 For light emitted from an ordinary lamp, so-called thermal light, the indeterminacy product on the left-hand side of the above inequality is a large number. Equivalently, the indeterminacy product for the action (12) is a large multiple of the quantum of action. For laser beams, i.e., beams of coherent light,** the indeterminacy product is close to 1/2. An illustration of coherent light is given in Figure 22. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Today it is possible to produce light for which the product of the two indeterminacies in equation (13) is near 1/2, but whose two values differ (in the units of the so-called Ref. 19 phasor space illustrated in Figure 21). Such light is called non-classical or squeezed. The photon statistics is either hyper- or sub-Poissonian. Such light beams require involved laboratory set-ups for their production and are used in many modern research applic- ations. Non-classical light has to be treated extremely carefully, as the smallest disturb- ances transforms it back into ordinary coherent (or even thermal light), in which Pois- son (or even Bose-Einstein) statistics hold again. A general overview of the main types of light beams is given in Figure 21, together with their intensity and phase behaviour. (Several properties shown in the figure are defined for a single phase space cell only.) * A large photon number is assumed in the expression. This is obvious, as Δ𝜑 cannot grow beyond all bounds, more precisely, not beyond 2π. The exact relations are ℏ Δ𝐼 Δ cos 𝜑 ⩾ |⟨sin 𝜑⟩| 2 ℏ Δ𝐼 Δ sin 𝜑 ⩾ |⟨cos 𝜑⟩| (11) 2 where ⟨𝑥⟩ denotes the expectation value of the observable 𝑥. ** Coherent light is light for which the photon number probability distribution is Poissonian; in particular, the variance is equal to the mean photon number. Coherent light is best described as composed of photons in coherent quantum states. Such a (canonical) coherent state, or Glauber state, is formally a state with Δ𝜑 → 1/𝑛 and Δ𝑛 → 𝑛. 2 light – and the quantum of action 49 Thermal equilibrium light Coherent laser light Non-classical, Non-classical, phase- squeezed light intensity-squeezed light Photon clicks show bunching Weak bunching Strong bunching Anti-bunching time Intensity I(t) time Photon number probability Bose-Einstein Poisson hyper- sub- (super-Poisson) Poisson Poisson Motion Mountain – The Adventure of Physics <n> n Intensity correlation g2(t) 2 bunching 2 weak bunching 2 strong bunching 2 anti-bunching 1 1 1 1 coherence time delay Phasor Im diagram ω ω ω ω copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Re F I G U R E 21 Four types of light and their photon properties: thermal light, laser light, and two extreme types of non-classical, squeezed light. One extreme of non-classical light is phase-squeezed light. Since a phase-squeezed light beam has an (almost) determined phase, the photon number in such a beam fluc- tuates from zero to (almost) infinity. In other words, in order to produce coherent laser light that approximates a pure sine wave as perfectly as possible, we must accept that the photon number is as undetermined as possible. Such a beam has extremely small phase fluctuations that provide high precision in interferometry; the phase noise is as low as possible. The other extreme of non-classical light is a beam with a given, fixed number of photons, and thus with an extremely high phase indeterminacy. In such an amplitude- squeezed light beam, the phase fluctuates erratically.* This sort of squeezed, non-classical * The most appropriate quantum states to describe such light are called number states, sometimes Fock states. These states are stationary, thus eigenstates of the Hamiltonian, and contain a fixed number of photons. 50 2 light – and the quantum of action 8 6 4 electric field (a. u.) 2 0 -2 -4 -6 -8 Motion Mountain – The Adventure of Physics 0 2 4 6 8 10 12 14 16 18 20 time (a. u.) F I G U R E 22 A simple way to illustrate the indeterminacy of a light beam’s intensity and phase: the measured electric ﬁeld of a coherent electromagnetic wave with low intensity, consisting of about a dozen photons. The cloudy sine wave corresponds to the phasor diagram at the bottom of the second column in the previous overview. For large number of photons, the relative noise amplitude is negligible. (© Rüdiger Paschotta) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net light is ideal for precision intensity measurements as it provides the lowest intensity noise available. This kind of light shows anti-bunching of photons. To gain more insight, sketch the graphs corresponding to Figure 22 for phase-squeezed and for amplitude-squeezed Challenge 36 s light. In contrast, the coherent light that is emitted by laser pointers and other lasers lies between the two extreme types of squeezed light: the phase and photon number inde- terminacies are of similar magnitude. The observations about thermal light, coherent laser light and non-classical light high- light an important property of nature: the number of photons in a light beam is not a well-defined quantity. In general, it is undetermined, and it fluctuates. Photons, unlike stones, cannot be counted precisely – as long as they are propagating and not absorbed. In flight, it is only possible to determine an approximate, average photon number, within the limits set by indeterminacy. Is it correct to claim that the number of photons at the Challenge 37 ny beginning of a beam is not necessarily the same as the number at the end of the beam? The fluctuations in the number of photons are of most importance at optical frequen- cies. At radio frequencies, the photon number fluctuations are usually negligible, due to the low photon energies and the usually high photon numbers involved. Conversely, at gamma-ray energies, wave effects play little role. For example, we saw that in deep, dark intergalactic space, far from any star, there are about 400 photons per cubic centimetre; they form the cosmic background radiation. This photon density number, like the num- ber of photons in a light beam, also has a measurement indeterminacy. Can you estimate Challenge 38 s it? 2 light – and the quantum of action 51 The Mach-Zehnder interferometer source detectors mirrors beam beam splitter splitter possible two identical light photons paths Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net F I G U R E 23 The Mach–Zehnder interferometer and a practical realization, about 0.5 m in size (© Félix Dieu and Gaël Osowiecki). In short, unlike pebbles, photons are countable, but their number is not fixed. And this is not the only difference between photons and pebbles. The positions of photons Where is a photon when it moves in a beam of light? Quantum theory gives a simple answer: nowhere in particular. This is proved most spectacularly by experiments with interferometers, such as the basic interferometer shown in Figure 23. Interferometers show that even a beam made of a single photon can be split, led along two different paths, and then recombined. The resulting interference shows that the single photon cannot be 52 2 light – and the quantum of action said to have taken either of the two paths. If one of the two paths is blocked, the pattern on the screen changes. In other words, somehow the photon must have taken both paths at the same time. Photons cannot be localized: they have no position.* We come to the conclusion that macroscopic light pulses have paths, but the indi- vidual photons in it do not. Photons have neither position nor paths. Only large numbers of photons can have positions and paths, and then only approximately. The impossibility of localizing photons can be quantified. Interference shows that it is impossible to localize photons in the direction transverse to the motion. It might seem less difficult to localize photons along the direction of motion, when it is part of a light pulse, but this is a mistake. The quantum of action implies that the indeterminacy in the longitudinal position is given at least by the wavelength of the light. Can you confirm Challenge 39 e this? It turns out that photons can only be localized within a coherence length. In fact, the transversal and the longitudinal coherence length differ in the general case. The longit- udinal coherence length (divided by 𝑐) is also called temporal coherence, or simply, the Page 49 coherence time. It is also indicated in Figure 21. The impossibility of localizing photons is Motion Mountain – The Adventure of Physics a consequence of the quantum of action. For example, the transverse coherence length is due to the indeterminacy of the transverse momentum; the action values for paths lead- ing to points separated by less than a coherence length differ by less than the quantum of action ℏ. Whenever a photon is detected somewhere, e.g., by absorption, a precise state- ment on its direction or its origin cannot be made. Sometimes, in special cases, there can be a high probability for a certain direction or source, though. Lack of localisation means that photons cannot be simply visualized as short wave trains. For example, we can increase the coherence length by sending light through a nar- row filter. Photons are truly unlocalizable entities, specific to the quantum world. Photons copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net are neither little stones nor little wave packets. Conversely, ‘light path’, ‘light pulse pos- ition’ and ‘coherence’ are properties of a photon ensemble, and do not apply to a single photon. Whenever photons can almost be localized along their direction of motion, as in co- herent light, we can ask how photons are lined up, one after the other, in a light beam. Of course, we have just seen that it does not make sense to speak of their precise position. But do photons in a perfect beam arrive at almost-regular intervals? To the shame of physicists, the study of photon correlations was initiated by two astro- nomers, Robert Hanbury Brown and Richard Twiss, in 1956, and met with several years Ref. 20 of disbelief. They varied the transversal distance of the two detectors shown in Figure 24 – from a few to 188 m – and measured the intensity correlations between them. Hanbury Brown and Twiss found that the intensity fluctuations within the volume of coherence are correlated. Thus the photons themselves are correlated. With this experiment, they were able to measure the diameter of numerous distant stars. Inspired by the success of Hanbury Brown and Twiss, researchers developed a simple method to measure the probability that a second photon in a light beam arrives at a given time after the first one. They simply split the beam, put one detector in the first branch, and varied the position of a second detector in the other branch. The set-up is sketched in Figure 25. Such an experiment is nowadays called a Hanbury Brown Twiss experiment. * We cannot avoid this conclusion by saying that photons are split at the beam splitter: if we place a detector in each arm, we find that they never detect a photon at the same time. Photons cannot be divided. 2 light – and the quantum of action 53 F I G U R E 24 The original experimental set-up with which Hanbury Brown and Twiss measured stellar diameters at Narrabri in Australia. The distance between the two light collectors could be changed by Motion Mountain – The Adventure of Physics moving them on rails. The light detectors are at the end of the poles and each of them, as they wrote, ‘collected light as rain in a bucket.’ (© John Davis). The Hanbury Brown–Twiss experiment light detector F I G U R E 25 How incoming D1 to measure light photon statistics beam with an copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net electronic intensity adjustable coincidence correlator or position counter coincidence counter, the variation being measured by light detector varying the D2 position of a detector. One finds that, for coherent light within the volume of coherence, the clicks in the two counters – and thus the photons themselves – are correlated. To be more precise, such experiments show that whenever the first photon hits, the second photon is most likely to hit just afterwards. Thus, photons in light beams are bunched. Bunching is one of the many results showing that photons are quantons, that they are indeed necessary to de- scribe light, and that they are unlocalizable entities. As we will see below, the result also Page 63 implies that photons are bosons. Every light beam has an upper time limit for bunching: the coherence time. For times longer than the coherence time, the probability for bunching is low, and independent of the time interval, as shown in Figure 25. The coherence time characterizes every light 54 2 light – and the quantum of action Ekin kinetic energy of emitted electrons lamp electrons Ekin=h (ω−ωt) threshold F I G U R E 26 The kinetic metal plate energy of electrons frequency of lamp light ω in vacuum emitted in the photoelectric effect. Motion Mountain – The Adventure of Physics beam. In fact, it is often easier to think in terms of the coherence length of a light beam. For thermal light, the coherence length is only a few micrometres: a small multiple of the wavelength. The largest coherence lengths, of over 300 000 km, are obtained with research lasers that have an extremely narrow laser bandwith of just 1 Hz. Interestingly, Ref. 21 coherent light is even found in nature: several special stars have been found to emit it. Although the intensity of a good laser beam is almost constant, the photons do not arrive at regular intervals. Even the best laser light shows bunching, though with dif- Page 49 ferent statistics and to a lesser degree than lamp light, as illustrated in Figure 21. Light whose photons arrive regularly, thus exhibiting so-called (photon) anti-bunching, is obvi- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ously non-classical in the sense defined above; such light can be produced only by special experimental arrangements. Extreme examples of this phenomenon are being investig- ated at present by several research groups aiming to construct light sources that emit one photon at a time, at regular time intervals, as reliably as possible. In short, we can state that the precise photon statistics in a light beam depends on the mechanism of the light source. In summary, experiments force us to conclude that light is made of photons, but also that photons cannot be localized in light beams. It makes no sense to talk about the position of a photon in general; the idea makes sense only in some special situations, and then only approximately and as a statistical average. Are photons necessary? In light of the results uncovered so far, the answer to the above question is obvious. But the issue is tricky. In textbooks, the photoelectric effect is usually cited as the first and most obvious experimental proof of the existence of photons. In 1887, Heinrich Hertz observed that for certain metals, such as lithium or caesium, incident ultraviolet light leads to charging of the metal. Later studies of the effect showed that the light causes emission of electrons, and that the energy of the ejected electrons does not depend on the intensity of the light, but only on the difference between ℏ times its frequency and a material-dependent threshold energy. Figure 26 summarizes the experiment and the measurements. 2 light – and the quantum of action 55 In classical physics, the photoelectric effect is difficult to explain. But in 1905, Albert Ref. 22 Einstein deduced the measurements from the assumption that light is made of photons of energy 𝐸 = ℏ𝜔. He imagined that this energy is used partly to take the electron over the threshold, and partly to give it kinetic energy. More photons only lead to more elec- trons, not to faster ones. In 1921, Einstein received the Nobel Prize for the explanation of the photoelectric effect. But Einstein was a genius: he deduced the correct result by a somewhat incorrect reasoning. The (small) mistake was the assumption that a classical, continuous light beam would produce a different effect. In fact, it is easy to see that a classical, continuous electromagnetic field interacting with discrete matter, made of dis- crete atoms containing discrete electrons, would lead to exactly the same result, as long as the motion of electrons is described by quantum theory. Several researchers confirmed Ref. 23 this early in the twentieth century. The photoelectric effect by itself does not imply the existence of photons. Indeed, many researchers in the past were unconvinced that the photoelectric effect shows the existence of photons. Historically, the most important argument for the neces- Motion Mountain – The Adventure of Physics sity of light quanta was given by Henri Poincaré. In 1911 and 1912, aged 57 and only a few months before his death, he published two influential papers proving that the radiation law of black bodies – in which the quantum of action had been discovered by Max Planck Ref. 24 – requires the existence of photons. He also showed that the amount of radiation emitted by a hot body is finite only because of the quantum nature of the processes leading to light emission. A description of these processes in terms of classical electrodynamics would lead to (almost) infinite amounts of radiated energy. Poincaré’s two influential papers convinced most physicists that it was worthwhile to study quantum phenomena in more detail. Poincaré did not know about the action limit 𝑆 ⩾ ℏ; yet his argument is based on copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net the observation that light of a given frequency has a minimum intensity, namely a single photon. Such a one-photon beam may be split into two beams, for example by using a half-silvered mirror. However, taken together, those two beams never contain more than a single photon. Another interesting experiment that requires photons is the observation of ‘molecules Ref. 25 of photons’. In 1995, Jacobson et al. predicted that the de Broglie wavelength of a packet of photons could be observed. According to quantum theory, the packet wavelength is given by the wavelength of a single photon divided by the number of photons in the packet. The team argued that the packet wavelength could be observable if such a packet could be split and recombined without destroying the cohesion within it. In 1999, this effect was indeed observed by de Pádua and his research group in Brazil. They used a careful set-up with a nonlinear crystal to create what they call a biphoton, and observed its interference properties, finding a reduction in the effective wavelength by the predicted factor of two. Ref. 26 Since then, packages with three and even four entangled photons have been created and observed. Yet another argument for the necessity of photons is the above-mentioned recoil felt Page 43 by atoms emitting light. The recoil measured in these cases is best explained by the emis- sion of a photon in a particular direction. In contrast, classical electrodynamics predicts the emission of a spherical wave, with no preferred direction. Page 49 Obviously, the observation of non-classical light, also called squeezed light, also argues for the existence of photons, as squeezed light proves that photons are indeed an intrinsic Ref. 27 aspect of light, necessary even when interactions with matter play no role. The same is 56 2 light – and the quantum of action lasers or other pocket lamps coherent light source F I G U R E 27 Two situations in which light crosses light: different light sources lead to different results. Motion Mountain – The Adventure of Physics true for the Hanbury Brown–Twiss effect. Finally, the spontaneous decay of excited atomic states also requires the existence of photons. This cannot be explained by a continuum description of light. In summary, the concept of a photon is indeed necessary for a precise description of light; but the details are often subtle, as the properties of photons are unusual and require a change in our habits of thought. To avoid these issues, most textbooks stop discussing photons after coming to the photoelectric effect. This is a pity, as it is only then that things get interesting. Ponder the following. Obviously, all electromagnetic fields are made of photons. At present, photons can be counted for gamma rays, X-rays, ultraviolet copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net light, visible light and infrared light. However, for lower frequencies, such as radio waves, photons have not yet been detected. Can you imagine what would be necessary to count Challenge 40 s the photons emitted from a radio station? This issue leads directly to the most important question of all: Interference: how can a wave be made up of particles? “ Die ganzen fünfzig Jahre bewusster Grübelei haben mich der Antwort auf die Frage ‘Was sind Lichtquanten?’ nicht näher gebracht. Heute glaubt zwar jeder Lump er wisse es, aber ” er täuscht sich. Albert Einstein, 1951 * If a light wave is made of particles, we must be able to explain each and every wave property in terms of photons. The experiments mentioned above already hint that this is possible only because photons are quantum particles. Let us take a more detailed look at this connection. Light can cross other light undisturbed, for example when the light beams from two pocket lamps shine through each other. This observation is not hard to explain with * ‘Fifty years of conscious brooding have not brought me nearer to the answer to the question ‘What are light quanta?’ Nowadays every bounder thinks he knows it, but he is wrong.’ Einstein wrote this a few years Ref. 28 before his death in a letter to Michele Besso. 2 light – and the quantum of action 57 Motion Mountain – The Adventure of Physics F I G U R E 28 Examples of interference patterns that appear when coherent light beams cross: the interference produced by a self-made parabolic telescope mirror of 27 cm diameter, and a speckle laser pattern on a rough surface (© Mel Bartels, Epzcaw). photons; since photons do not interact with each other, and are point-like, they ‘never’ hit each other. In fact, there is an extremely small positive probability for their interac- Vol. V, page 130 tion, as we will find out later, but this effect is not observable in everyday life. But if two coherent light beams, i.e., two light beams of identical frequency and fixed phase relation cross, we observe alternating bright and dark regions: so-called interfer- ence fringes. The schematic set-up is shown in Figure 27. Examples of actual interference copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net effects are given in Figure 28 and Figure 29. How do these interference fringes appear?* How can it be that photons are not detected in the dark regions? We already know the only possible answer: the brightness at a given place corresponds to the probability that a photon will arrive there. The fringes imply: ⊳ Photons behave like moving little arrows. Some further thought leads to the following description: — The arrow is always perpendicular to the direction of motion. — The arrow’s direction stays fixed in space when the photons move. — The length of an arrow shrinks with the square of the distance travelled. — The probability of a photon arriving somewhere is given by the square of an arrow. — The final arrow is the sum of all the arrows arriving there by all possible paths. — Photons emitted by single-coloured sources are emitted with arrows of constant length pointing in the direction 𝜔𝑡; in other words, such sources spit out photons with a rotating mouth. — Photons emitted by incoherent sources – e.g., thermal sources, such as pocket lamps – are emitted with arrows of constant length pointing in random directions. * If lasers are used, fringes can only be observed if the two beams are derived from a single beam by splitting, Challenge 41 s or if two expensive high-precision lasers are used. (Why?) 58 2 light – and the quantum of action Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net F I G U R E 29 Top: calculated interference patterns – and indistinguishable from observed ones under ideal, “textbook” conditions – produced by two parallel narrow slits illuminated with green light and with white light. Bottom: two Gaussian beams interfering at an angle (© Dietrich Zawischa, Rüdiger Paschotta). With this simple model* we can explain the wave behaviour of light. In particular, we can describe the interference stripes seen in laser experiments, as shown schematically in Figure 30. You can check that in some regions the two arrows travelling through the two slits add up to zero for all times. No photons are detected there: those regions are black. In other regions, the arrows always add up to the maximal value. These regions are always bright. Regions in between have intermediate shades. Obviously, in the case of usual pocket lamps, shown in the left-hand diagram of Figure 27, the brightness in the common region also behaves as expected: the averages simply add up. Obviously, the photon model implies that an interference pattern is built up as the sum of a large number of single-photon hits. Using low-intensity beams, we should therefore be able to see how these little spots slowly build up an interference pattern by accumu- lating in the bright regions and never hitting the dark regions. This is indeed the case, as * The model gives a correct description of light except that it neglects polarization. To add polarization, it is necessary to combine arrows that rotate in both senses around the direction of motion. 2 light – and the quantum of action 59 two lasers or screen point sources S1 s d S2 the arrow model: t1 Motion Mountain – The Adventure of Physics t2 t3 F I G U R E 30 Interference and the description of light with arrows (at three instants of time). Page 43 we have seen earlier on. All experiments confirm this description. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net In other words, interference is the superposition of coherent light fields or, more gen- erally, of coherent electromagnetic fields. Coherent light fields have specific, more regu- lar photon behaviour, than incoherent light fields. We will explore the details of photon statistics in more detail shortly. In summary, photons are quantum particles. Quantum particles can produce interfer- ence patterns – and all other wave effects – when they appear in large numbers, because they are described by an arrow whose length squared gives the probability for its detec- tion. Interference of a single photon It is important to point out that interference between two light beams is not the result of two different photons cancelling each other out or being added together. Such cancella- tion would contradict conservation of energy and momentum. Interference is an effect applicable to each photon separately – as shown in the previous section – because each photon is spread out over the whole set-up: each photon takes all possible paths. As Paul Ref. 29 Dirac stressed: ⊳ Each photon interferes only with itself. Interference of a photon with itself only occurs because photons are quantons, and not classical particles. 60 2 light – and the quantum of action screen source image mirror F I G U R E 31 Light reﬂected by a arrow sum mirror, and the corresponding Motion Mountain – The Adventure of Physics arrows (at an instant of time). Dirac’s oft-quoted statement leads to a famous paradox: if a photon can interfere only with itself, how can two laser beams from two different lasers interfere with each other? The answer given by quantum physics is simple but strange: in the region where the Page 51 beams interfere – as mentioned above – it is impossible to say from which source a photon has come. The photons in the crossing region cannot be said to come from a specific source. Photons, also in the interference region, are quantons, and they indeed copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net interfere only with themselves. Another description of the situation is the following: ⊳ A photon interferes only within its volume of coherence. And in that volume, it is impossible to distinguish photons. In the coherence volume formed by the longitudinal and transversal coherence length – sometimes also called a phase space cell – we cannot completely say that light is a flow of photons, because a flow cannot be defined in it. Despite regular claims to the con- Page 66 trary, Dirac’s statement is correct, as we will see below. It is a strange consequence of the quantum of action. R eflection and diffraction deduced from photon arrows Waves also show diffraction. Diffraction is the change of propagation direction of light or any other wave near edges. To understand this phenomenon with photons, let us start with a simple mirror, and study reflection first. Photons (like all quantum particles) move from source to detector by all possible paths. As Richard Feynman,* who discovered this explanation, liked to stress, the term ‘all’ has to be taken literally. This is not a big deal in * Richard (‘Dick’) Phillips Feynman (b. 1918 New York City, d. 1988 Los Angeles), physicist, was one of the founders of quantum electrodynamics. He also discovered the ‘sum-over-histories’ reformula- tion of quantum theory, made important contributions to the theory of the weak interaction and to 2 light – and the quantum of action 61 source point arrow sum at point usual vanishes mirror screen source image Motion Mountain – The Adventure of Physics arrow sum at image striped mirror F I G U R E 32 The light reﬂected by a badly-placed mirror and by a grating. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net the explanation of interference. But in order to understand a mirror, we have to include all possibilities, however crazy they seem, as shown in Figure 31. As stated above, a light source emits rotating arrows. To determine the probability that light arrives at a certain location within the image, we have to add up all the arrows arriving at the same time at that location. For each path, the arrow orientation at the image is shown – for convenience only – below the corresponding segment of the mirror. The angle and length of the arriving arrow depends on the path. Note that the sum of all the arrows does not vanish: light does indeed arrive at the image. Moreover, the largest contribution comes from the paths near to the middle. If we were to perform the same calculation for another image location, (almost) no light would get there. In short, the rule that reflection occurs with the incoming angle equal to the outgoing angle is an approximation, following from the arrow model of light. In fact, a detailed quantum gravity, and co-authored a famous textbook, the Feynman Lectures on Physics, now online at www. feynmanlectures.info. He is one of those theoretical physicists who made his career mainly by performing complex calculations – but he backtracked with age, most successfully in his teachings and physics books, which are all worth reading. He was deeply dedicated to physics and to enlarging knowledge, and was a collector of surprising physical explanations. He helped building the nuclear bomb, wrote papers in top- less bars, avoided to take any professional responsibility, and was famously arrogant and disrespectful of authority. He wrote several popular books on the events of his life. Though he tried to surpass the genius of Wolfgang Pauli throughout his life, he failed in this endeavour. He shared the 1965 Nobel Prize in Physics for his work on quantum electrodynamics. 62 2 light – and the quantum of action light beam air water F I G U R E 33 If light were made of little stones, they would move faster in water. Motion Mountain – The Adventure of Physics calculation, with more arrows, shows that the approximation is quite precise: the errors are much smaller than the wavelength of the light. The proof that light does indeed take all these strange paths is given by a more spe- cialized mirror. As show in Figure 32, we can repeat the experiment with a mirror that reflects only along certain stripes. In this case, the stripes have been carefully chosen so that the corresponding path lengths lead to arrows with a bias in one direction, namely to the left. The arrow addition now shows that such a specialized mirror – usually called a grating – allows light to be reflected in unusual directions. Indeed, this behaviour is standard for waves: it is called diffraction. In short, the arrow model for photons allows copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net us to describe this wave property of light, provided that photons follow the ‘crazy’ prob- ability scheme. Do not get upset! As was said above, quantum theory is the theory for crazy people. You may wish to check that the arrow model, with the approximations it generates by summing over all possible paths, automatically ensures that the quantum of action is Challenge 42 s indeed the smallest action that can be observed. R efraction and partial reflection from photon arrows All waves have a signal velocity. The signal velocity also depends on the medium in which they propagate. As a consequence, waves show refraction when they move from one me- dium into another with different signal velocity. Interestingly, the naive particle picture of photons as little stones would imply that light is faster in materials with high refractive Challenge 43 e indices: the so-called dense materials. (See Figure 33.) Can you confirm this? However, experiments show that light in dense materials moves slowly. The wave picture has no Challenge 44 e difficulty explaining this observation. (Can you confirm this?) Historically, this was one of the arguments against the particle theory of light. In contrast, the arrow model of light Challenge 45 e presented above is able to explain refraction properly. It is not difficult: try it. Waves also reflect partially from materials such as glass. This is one of the most dif- ficult wave properties to explain with photons. But it is one of the few effects that is not explained by a classical wave theory of light. However, it is explained by the arrow model, as we will find out. Partial reflection confirms the first two rules of the arrow model. Par- 2 light – and the quantum of action 63 Page 57 tial reflection shows that photons indeed behave randomly: some are reflected and other are not, without any selection criterion. The distinction is purely statistical. More about this issue shortly. From photons to waves In waves, the fields oscillate in time and space. One way to show how waves can be made of particles is to show how to build up a sine wave using a large number of photons. A Ref. 30 sine wave is a coherent state of light. The way to build them up was explained in detail by Roy Glauber. In fact, to build a pure sine wave, we need a superposition of a beam with one photon, a beam with two photons, a beam with three photons, and so on. Together, they give a perfect sine wave. As expected, its photon number fluctuates to the highest possible degree. If we repeat the calculation for non-ideal beams, we find that the indeterminacy rela- tion for energy and time is respected: every emitted beam will possess a certain spectral width. Purely monochromatic light does not exist. Similarly, no system that emits a wave Motion Mountain – The Adventure of Physics at random can produce a monochromatic wave. All experiments confirm these results. In addition, waves can be polarized. So far, we have disregarded this property. In the photon picture, polarization is the result of carefully superposing beams of photons spin- ning clockwise and anticlockwise. Indeed, we know that linear polarization can be seen as a result of superposing circularly-polarized light of both signs, using the proper phase. What seemed a curiosity in classical optics turns out to be a fundamental justification for quantum theory. Finally, photons are indistinguishable. When two photons of the same colour cross, there is no way to say afterwards which of the two is which. The quantum of action makes copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net this impossible. The indistinguishability of photons has an interesting consequence. It is impossible to say which emitted photon corresponds to which arriving photon. In other words, there is no way to follow the path of a photon, as we are used to following the Page 52 path of a billiard ball. Photons are indeed indistinguishable. In addition, the experiment Ref. 31 by Hanbury Brown and Twiss implies that photons are bosons. We will discover more Page 112 details about the specific indistinguishability of bosons later on. In summary, we find that light waves can indeed be described as being built of particles. However, this is only correct with the proviso that photons — are not precisely countable – never with a precision better than √𝑁 , — are not localizable – never with a precision better than the coherence length, — have no size, no charge and no (rest) mass, — show a phase that increases as 𝜔𝑡, i.e., as the product of frequency and time, — carry spin 1, — of the same frequency are indistinguishable bosons – within a coherence volume, — can take any path whatsoever – as long as allowed by the boundary conditions, — have no discernable origin, and — have a detection probability given by the square of the sum of amplitudes* for all allowed paths leading to the point of detection. * The amplitude of a photon field, however, cannot and should not be identified with the wave function of any massive spin 1 particle. 64 2 light – and the quantum of action In other words, light can be described as made of particles only if these particles have special, quantum properties. These quantum properties differ from everyday particles and allow photons to behave like waves whenever they are present in large numbers. C an light move faster than light? – R eal and virtual photons In a vacuum, light can move faster than 𝑐, as well as slower than 𝑐. The quantum principle provides the details. As long as this principle is obeyed, the speed of a short light flash can differ – though only by a tiny amount – from the ‘official’ value. Can you estimate Challenge 46 ny the allowable difference in arrival time for a light flash coming from the dawn of time? The arrow description for photons gives the same result. If we take into account the crazy possibility that photons can move with any speed, we find that all speeds very dif- ferent from 𝑐 cancel out. The only variation that remains, translated into distances, is the Challenge 47 ny indeterminacy of about one wavelength in the longitudinal direction, which we men- tioned above. In short, light, or real photons, can indeed move faster than light, though only by an Motion Mountain – The Adventure of Physics amount allowed by the quantum of action. For everyday situations, i.e., for high values of the action, all quantum effects average out, including light and photon velocities different from 𝑐. Ref. 32 Not only the position, but also the energy of a single photon can be undefined. For example, certain materials split one photon of energy ℏ𝜔 into two photons, whose two energies add up to the original one. Quantum mechanics implies that the energy parti- tioning is known only when the energy of one of the two photons is measured. Only at that very instant is the energy of the second photon known. Before the measurement, both photons have undefined energies. The process of energy fixing takes place instant- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Page 153 aneously, even if the second photon is far away. We will explain below the background to this and similar strange effects, which seem to be faster than light. In fact, despite the appearance, these observations do not involve faster-than-light transmission of energy Challenge 48 s or information. More bizarre consequences of the quantum of action appear when we study static elec- tric fields, such as the field around a charged metal sphere. Obviously, such a field must also be made of photons. How do they move? It turns out that static electric fields are made of virtual photons. Virtual photons are photons that do not appear as free particles: they only appear for an extremely short time before they disappear again. In the case of a static electric field, they are longitudinally polarized, and do not carry energy away. Virtual photons, like other virtual particles, are ‘shadows’ of particles that obey Δ𝑥Δ𝑝 ⩽ ℏ/2 . (14) Rather than obeying the usual indeterminacy relation, they obey the opposite relation, which expresses their very brief appearance. Despite their intrinsically short life, and des- pite the impossibility of detecting them directly, virtual particles have important effects. Page 193 We will explore them in detail shortly. In fact, the vector potential 𝐴 allows four polarizations, corresponding to the four coordinates (𝑡, 𝑥, 𝑦, 𝑧). It turns out that for the photons one usually talks about – the free or real photons – the polarizations in the 𝑡 and 𝑧 directions cancel out, so that one 2 light – and the quantum of action 65 observes only the 𝑥 and 𝑦 polarizations in actual experiments. For bound or virtual photons, the situation is different. All four polarizations are pos- sible. Indeed, the z and t polarizations of virtual photons – which do not appear for real photons, i.e., for free photons – are the ones that can be said to be the building blocks of static electric and magnetic fields. In other words, static electric and magnetic fields are continuous flows of virtual photons. In contrast to real photons, virtual photons can have mass, can have spin dir- ections not pointing along the path of motion, and can have momentum opposite to their direction of motion. Exchange of virtual photons leads to the attraction of bodies of different charge. In fact, virtual photons necessarily appear in any description of elec- Vol. V, page 122 tromagnetic interactions. Later on we will discuss their effects further – including the famous attraction of neutral bodies. Vol. II, page 72 We have seen already early on that virtual photons, for example those that are needed to describe collisions of charges, must be able to move with speeds higher than that of light. This description is required in order to ensure that the speed of light remains a Motion Mountain – The Adventure of Physics limit in all experiments. In summary, it might be intriguing to note that virtual photons, in contrast to real photons, are not bound by the speed of light; but it is also fair to say that virtual photons move faster-than-light only in a formal sense. Indeterminacy of electric fields We have seen that the quantum of action implies an indeterminacy for light intensity. Since light is an electromagnetic wave, this indeterminacy implies similar, separate limits for electric and magnetic fields at a given point in space. This conclusion was first drawn copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Ref. 33 in 1933 by Bohr and Rosenfeld. They started from the effects of the fields on a test particle of mass 𝑚 and charge 𝑞, which are described by: 𝑚𝑎 = 𝑞 (𝐸 + 𝑣 × 𝑏) . (15) Since it is impossible to measure both the momentum and the position of a particle, they Challenge 49 ny deduced an indeterminacy for the electrical field, given by ℏ Δ𝐸 = , (16) 𝑞 Δ𝑥 𝑡 where 𝑡 is the measurement time and Δ𝑥 is the position indeterminacy. Thus every value of an electric field, and similarly of a magnetic field, possesses an indeterminacy. The state of the electromagnetic field behaves like the state of matter in this respect: both follow an indeterminacy relation. How can virtual photon exchange lead to at traction? Exchange of real photons always leads to recoil. But exchange of virtual photons can lead either to attraction or repulsion, depending on the signs of the two charges involved. This is worth looking at. 66 2 light – and the quantum of action We start with two localized charges of same sign, located both on the 𝑥-axis, and want to determine the momentum transferred from the charge on the right side via a virtual photon to the charge on the left side. For the virtual photon, the important part of its state in momentum space is its imagin- ary part, which, if emitted by a negative charge, has a positive peak (delta function shape) at the negative of its momentum value and a negative peak at its positive momentum value. When the virtual photon hits the other charged particle, on the left, it can push it either to the left or to the right. The probability amplitude for each process is given by the particle charge times the photon momentum value times 𝑖 times time. Both amplitudes need to be added. In the case that the second particle has the same charge as the first, the effect of the virtual photon absorption in momentum space is to add a wave function that originally was antisymmetric and positively valued on the positive axis, and that is then shifted to the left, to a second wave function which originally was the negative of the first, but Motion Mountain – The Adventure of Physics is then shifted to the right. The result for this one-photon absorption process is a real- valued, antisymmetric function in momentum space, with positive values for negative momenta, and negative values for positive momenta. To understand repulsion, we need to add the wave function for this one-photon pro- cess to the zero-photon (thus unmodified) function of the second particle, and then square the sum. This unmodified function was positive in the case of same charges. The squaring process of the sum yields a probability distribution in momentum space whose maximum is at a negative momentum value; thus the second particle has been repelled from the first. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net If the charges had different signs, the maximum of the sum would be at a positive mo- mentum value, and the second particle would be attracted to the first. In short, attraction or repulsion is determined by the interference between the wave function for one-photon absorption (more precisely, for odd-photon-number absorption) and the wave function for zero-photon absorption (more precisely, for even-photon-number absorption). C an t wo photons interfere? Page 59 In 1930, Paul Dirac made the famous statement already mentioned earlier on: ⊳ Each photon interferes only with itself. Interference between two different Ref. 29 photons never occurs. Often this statement is misinterpreted as implying that light from two separate photon sources cannot interfere. Unfortunately, this false interpretation has spread through a part Ref. 34 of the literature. Everybody can check that this statement is incorrect with a radio: signals from two distant radio stations transmitting on the same frequency lead to beats in amp- litude, i.e., to wave interference. (This should not to be confused with the more common radio interference, which usually is simply a superposition of intensities.) Radio trans- mitters are coherent sources of photons, and any radio receiver shows that signals form two different sources can indeed interfere. In 1949, interference of fieds emitted from two different photon sources has been 2 light – and the quantum of action 67 demonstrated also with microwave beams. From the nineteen fifties onwards, numerous experiments with two lasers and even with two thermal light sources have shown light Ref. 35 interference. For example, in 1963, Magyar and Mandel used two ruby lasers emitting light pulses and a rapid shutter camera to produce spatial interference fringes. However, all these experimental results with two interfering sources do not contradict the statement by Dirac. Indeed, two photons cannot interfere for several reasons. — Interference is a result of the space-time propagation of waves; photons appear only when the energy–momentum picture is used, mainly when interaction with matter takes place. The description of space-time propagation and the particle picture are mutually exclusive – this is one aspect of the complementary principle. Why does Dirac seem to mix the two in his statement? Dirac employs the term ‘photon’ in a very general sense, as quantized state of the electromagnetic field. When two coher- ent beams are superposed, the quantized entities, the photons, cannot be ascribed to either of the sources. Interference results from superposition of two coherent states, not of two particles. Motion Mountain – The Adventure of Physics — Interference is only possible if one cannot know where the detected photon comes from. The quantum mechanical description of the field in a situation of interference never allows ascribing photons of the superposed field to one of the sources. In other words, if it is possible to say from which source a detected photon comes from, in- terference cannot be observed. — Interference between two coherent beams requires a correlated or fixed phase between them, i.e., an undetermined particle number; in other words, interference is possible if and only if the photon number for each of the two beams is unknown. And a beam has an unknown photon number when the number indeterminacy is of copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net similar size as the average number. The statement of Dirac thus depends on the definition of the term ‘photon’. A better choice of words is to say that interference is always between two (indistinguishable) state histories, but never between two quantum particles. Or, as expressed above: ⊳ A photon interferes only within its volume of coherence, i.e., within its own cell of phase space. Outside, there is no interference. And inside that volume or cell, it is impossible to distinguish photons, states or histories. The concept of ‘photon’ remains deep even today. The quantum particle model of coher- ence and light remains fascinating to this day. Summarizing, we can say: Two different electromagnetic beams can interfere, but two different photons cannot. Curiosities and fun challenges ab ou t photons Can one explain refraction with photons? Newton was not able to do so, but today we can. In refraction by a horizontal surface, as shown in Figure 34, the situation is transla- tionally invariant along the horizontal direction. Therefore, the momentum component along this direction is conserved: 𝑝1 sin 𝛼1 = 𝑝2 sin 𝛼2 . The photon energy 𝐸 = 𝐸1 = 𝐸2 is obviously conserved. The index of refraction 𝑛 is defined in terms of momentum and 68 2 light – and the quantum of action p1 α1 air water p2 α2 F I G U R E 34 Refraction and photons. energy as 𝑐𝑝 Motion Mountain – The Adventure of Physics 𝑛= . (17) 𝐸 Challenge 50 e The ‘law’ of refraction follows: sin 𝛼1 =𝑛. (18) sin 𝛼2 The relation is known since the middle ages. There is an important issue here. In a material, the velocity of a photon 𝑣 = 𝛿𝐸/𝛿𝑝 in a light ray differs from the phase velocity 𝑢 = 𝐸/𝑝 that enters into the calculation. In copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net summary, inside matter, the concept of photon must be used with extreme care. ∗∗ If an electromagnetic wave has amplitude 𝐴, the photon density 𝑑 is 𝐴2 𝑑= . (19) ℏ𝜔 Challenge 51 ny Can you show this? ∗∗ Show that for a laser pulse in vacuum, the coherence volume increases during propaga- Challenge 52 e tion, whereas the volume occupied in phase space remains constant. Its entropy is con- stant, as its path is reversible. ∗∗ A typical effect of the quantum ‘laws’ is the yellow colour of the lamps used for street illumination in most cities. They emit pure yellow light of (almost) a single frequency; that is why no other colours can be distinguished in their light. According to classical electrodynamics, harmonics of that light frequency should also be emitted. Experiments show, however, that this is not the case; classical electrodynamics is thus wrong. Is this Challenge 53 s argument correct? 2 light – and the quantum of action 69 Motion Mountain – The Adventure of Physics F I G U R E 35 The blue shades of the sky and the colours of clouds are due to various degrees of Rayleigh, Mie and Tyndall scattering (© Giorgio di Iorio). ∗∗ How can you check whether a single-photon-triggered bomb is functional without ex- ploding it? This famous puzzle, posed by Avshalom Elitzur and Lev Vaidman, requires Challenge 54 ny interference for its solution. Can you find a way? copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ∗∗ What happens to photons that hit an object but are not absorbed or transmitted? Gener- ally speaking, they are scattered. Scattering is the name for any process that changes the motion of light (or that of any other wave). The details of the scattering process depend on the object; some scattering processes only change the direction of motion, others also change the frequency. Table 3 gives an overview of processes that scatter light. All scattering properties depend on the material that produces the deflection of light. Among others, the study of scattering processes explains many colours of transparent Page 171 materials, as we will see below. Challenge 55 e We note that the bending of light due to gravity is not called scattering. Why not? A summary on light : particle and wave In summary, light is a stream of light quanta or photons. A single photon is the smallest possible light intensity of a given colour. Photons, like all quantons, are quite different from everyday particles. In fact, we can argue that the only (classical) particle aspects of photons are their quantized energy, momentum and spin. In all other respects, photons are not like little stones. Photons move with the speed of light. Photons cannot be local- ized in light beams. Photons are indistinguishable. Photons are bosons. Photons have no mass, no charge and no size. It is more accurate to say that photons are calculating devices Ref. 36 to precisely describe observations about light. 70 2 light – and the quantum of action TA B L E 3 Types of light scattering. S c at t e r i n g S c at t e r e r D e ta i l s Examples type Rayleigh scattering atoms, molecules elastic, intensity blue sky, red evening changes as 1/𝜆4 , sky, blue cigarette scatterers smaller smoke than 𝜆/10 Mie scattering transparent objects, elastic, intensity blue sky, red droplets changes as 1/𝜆0.5 to evenings, blue 1/𝜆2 , scatterer size distant mountains around 𝜆 Geometric scattering edges elastic, scatterer sizebetter called larger than 𝜆 diffraction, used in interference Tyndall scattering non-transparent objects elastic, angle weakly smog, white clouds, Motion Mountain – The Adventure of Physics or not wavelength- fog, white cigarette dependent smoke Smekal–Raman excited atoms, molecules inelastic, light gains used in lidar scattering energy investigations of the atmosphere Inverse Raman atoms, molecules inelastic, light loses used in material scattering energy research Thomson scattering electrons elastic used for electron density copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net determination Compton scattering electrons inelastic, X-ray lose proves particle energy nature of light (see page 46) Brillouin scattering acoustic phonons, density inelastic, frequency used to study variations in solids/fluids shift of a few GHz phonons and to diagnose optical fibres Von Laue or X-ray crystalline solids elastic, due to used to determine scattering interference at crystal structures; crystal planes also called Bragg diffraction The strange properties of photons are the reason why earlier attempts to describe light as a stream of (classical) particles, such as the attempt of Newton, failed miserably, and were rightly ridiculed by other scientists. Indeed, Newton upheld his theory against all experimental evidence – especially with regard to light’s wave properties – which is something that a physicist should never do. Only after people had accepted that light is a wave, and then discovered and understood that quantum particles are fundamentally different from classical particles, was the quanton description successful. The quantum of action implies that all waves are streams of quantons. In fact, all waves 2 light – and the quantum of action 71 are correlated streams of quantons. This is true for light, for any other form of radiation, and for all forms of matter waves. The indeterminacy relations show that even a single quanton can be regarded as a wave; however, whenever it interacts with the rest of the world, it behaves as a particle. In fact, it is essential that all waves be made of quantons: if they were not, then interactions would be non-local, and objects could not be localized at all, contrary to experience. To decide whether the wave or the particle description is more appropriate, we can use the following criterion. Whenever matter and light interact, it is more appropriate to describe electromagnetic radiation as a wave if the wavelength 𝜆 satisfies ℏ𝑐 𝜆≫ , (20) 𝑘𝑇 where 𝑘 = 1.4 ⋅ 10−23 J/K is Boltzmann’s constant and 𝑇 is the temperature of the particle. If the wavelength is much smaller than the quantity on the right-hand side, the Motion Mountain – The Adventure of Physics particle description is most appropriate. If the two sides are of the same order of mag- Challenge 56 e nitude, both descriptions play a role. Can you explain the criterion? copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Chapter 3 MOT ION OF M AT T E R – BEYON D C L A S SIC A L PH YSIC S “ ” All great things begin as blasphemies. George Bernard Shaw T he existence of a smallest action has numerous important consequences for Motion Mountain – The Adventure of Physics he motion of matter. We start with a few experimental results that show hat the quantum of action is indeed the smallest measurable action value, also in the case of matter. Then we show that the quantum of action implies the existence of a phase and thus of the wave properties of matter. Finally, from the quantum of action, we deduce for the motion of matter the same description that we already found for light: matter particles behave like rotating arrows. Wine glasses, pencils and atoms – no rest “ ” Otium cum dignitate.** copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Cicero, De oratore. If the quantum of action is the smallest observable change in a physical system, then two observations of the same system must always differ. Thus there cannot be perfect rest in nature. Is that true? Experiments show that this is indeed the case. A simple consequence of the lack of perfect rest is the impossibility of completely filling a glass of wine. If we call a glass at maximum capacity (including surface tension effects, to make the argument precise) ‘full’, we immediately see that the situation re- quires the liquid’s surface to be completely at rest. This is never observed. Indeed, a com- pletely quiet surface would admit two successive observations that differ by less than ℏ. We could try to reduce all motions by reducing the temperature of the system. To achieve absolute rest we would need to reach absolute zero temperature. Experiments show that this is impossible. (Indeed, this impossibility, the so-called third ‘law’ of thermodynam- ics, is equivalent to the existence of a minimum action.) All experiments confirm: There is no rest in nature. In other words, the quantum of action proves the old truth that a glass of wine is always partially empty and partially full. The absence of microscopic rest, predicted by the quantum of action, is confirmed in many experiments. For example, a pencil standing on its tip cannot remain vertical, as shown in Figure 36, even if it is isolated from all disturbances, such as vibrations, air molecules and thermal motion. This – admittedly very academic – conclusion follows ** ‘Rest with dignity.’ 3 motion of matter – beyond classical physics 73 𝛼 axis F I G U R E 36 A falling pencil. from the indeterminacy relation. In fact, it is even possible to calculate the time after Challenge 57 d which a pencil must have fallen over. In practice however, pencils fall over much earlier, Motion Mountain – The Adventure of Physics because in usual conditions, external disturbances are much larger than the effects of the quantum of action. But the most important consequence of the absence of rest is another. The absence of rest for the electrons inside atoms prevents them from falling into the nuclei, despite their mutual attraction. In other words, the existence and the size of atoms, and thus of all matter, is a direct consequence of the absence of microscopic rest! We will explore Page 79 this consequence in more detail below. Since we are made of atoms, we can say: we only exist and live because of the quantum of action. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net No infinite measurement precision Not only does the quantum of action prevent the existence of rest; the quantum of ac- tion also prevents the observation or measurement of rest. In order to check whether an object is at rest, we need to observe its position with high precision. Because of the wave properties of light, we need a high-energy photon: only a high-energy photon has a small wavelength and thus allows a precise position measurement. As a result of this high energy, however, the object is disturbed. Worse, the disturbance itself is not pre- cisely measurable; so there is no way to determine the original position even by taking the disturbance into account. In short, perfect rest cannot be observed – even if it existed. Indeed, all experiments in which systems have been observed with highest precision confirm that perfect rest does not exist. The absence of rest has been confirmed for elec- trons, neutrons, protons, ions, atoms, molecules, atomic condensates and crystals. The absence of rest has been even confirmed for objects with a mass of about a tonne, as used in certain gravitational wave detectors. No object is ever at rest. The same argument on measurement limitations also shows that no measurement, of any observable, can ever be performed to infinite precision. This is another of the far- reaching consequences of the quantum of action. C o ol gas The quantum of action implies that rest is impossible in nature. In fact, even at extremely low temperatures, all particles inside matter are in motion. This fundamental lack of rest 74 3 motion of matter – beyond classical physics is said to be due to the so-called zero-point fluctuations. A good example is provided by the recent measurements of Bose–Einstein condensates. They are trapped gases, with a small number of atoms (between ten and a few million), cooled to extremely low tem- peratures (around 1 nK). The traps allow to keep the atoms suspended in mid-vacuum. These cool and trapped gases can be observed with high precision. Using elaborate ex- perimental techniques, Bose–Einstein condensates can be put into states for which Δ𝑝Δ𝑥 is almost exactly equal to ℏ/2 – though never lower than this value. These experiments confirm directly that there is no observable rest, but a fundamental fuzziness in nature. And the fuzziness is described by the quantum of action. This leads to an interesting puzzle. In a normal object, the distance between the atoms Challenge 58 s is much larger than their de Broglie wavelength. (Can you confirm this?) But today it is possible to cool objects to extremely low temperatures. At sufficiently low temperatures, Ref. 37 less than 1 nK, the wavelength of the atoms may be larger than their separation. Can you Challenge 59 s imagine what happens in such cases? Motion Mountain – The Adventure of Physics Flows and the quantization of mat ter “ ” Die Bewegung ist die Daseinsform der Materie. Friedrich Engels, Anti-Dühring.* Not only does the quantum of action make rest impossible, it also makes impossible any situation that does not change in time. The most important examples of (apparently) sta- tionary situations are flows. The quantum of action implies that no flow can be stationary. More precisely, a smallest action implies that no flow can be continuous. All flows fluc- tuate. In nature, all flows are made of smallest entities: all flows are made of quantum copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net particles. We saw above that this is valid for light; it also applies to matter flows. Two simple kinds of flow from our everyday experience directly confirm this consequence from the quantum of action: flows of fluids and flows of electricity. Fluid flows and quantons The flow of matter also exhibits smallest units. We mentioned early on in our adventure Vol. I, page 391 that a consequence of the particulate structure of liquids is that oil or any other smooth liquid produces noise when it flows through even the smoothest of pipes. We mentioned that the noise we hear in our ears in situations of absolute silence – for example, in a snowy and windless landscape in the mountains or in an anechoic chamber – is partly due to the granularity of blood flow in the veins. All experiments confirm that all flows of matter produce vibrations. This is a consequence of the quantum of action, and of the resulting granularity of matter. In fact, the quantum of action can be determined from noise measurements in fluids. Kno cking tables and quantized conductivit y If electrical current were a continuous flow, it would be possible to observe action values as small as desired. The simplest counter-example was discovered in 1996, by José Costa- Ref. 38, Ref. 39 Krämer and his colleagues. They put two metal wires on top of each other on a kitchen Ref. 12 * ‘Motion is matter’s way of being.’ 3 motion of matter – beyond classical physics 75 Motion Mountain – The Adventure of Physics F I G U R E 37 Steps in the ﬂow of electricity in metal wire crossings: the set-up, the nanowires at the basis of the effect, and three measurement results (© José Costa-Krämer, AAPT from Ref. 39). copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net table and attached a battery, a current-voltage converter – or simply a resistor – and a storage oscilloscope to them. Then they measured the electrical current while knocking on the table. That is all. Knocking the table breaks the contact between the two wires. In the last millisecond before the wires detach, the conductivity and thus the electrical current diminishes in regular steps of about 7 μA, as can easily be seen on the oscilloscope. Figure 37 shows such a measurement. This simple experiment could have beaten, if it had been performed a few years earlier, a number of other, enormously expensive experiments which dis- covered this same quantization at costs of several million euro each, using complex set- ups at extremely low temperatures. In fact, the quantization of conductivity appears in any electrical contact with a small cross-section. In such situations the quantum of action implies that the conductivity can Challenge 60 e only be a multiple of 2𝑒2 /ℏ ≈ (12 906 Ω)−1 . Can you confirm this result? Note that elec- trical conductivity can be as small as required; only the quantized electrical conductivity has the minimum value of 2𝑒2 /ℏ. Many more elaborate experiments confirm the observation of conductance steps. They force us to conclude that there is a smallest electric charge in nature. This smallest charge has the same value as the charge of an electron. Indeed, electrons turn out to be part of every atom, in a construction to be explained shortly. In metals, a large number of electrons can move freely: that is why metals conduct electricity so well and work as 76 3 motion of matter – beyond classical physics F I G U R E 38 Electrons beams diffract and interfere at multiple slits (© Claus Jönsson). Motion Mountain – The Adventure of Physics mirrors. In short, matter and electricity flow in smallest units. Depending on the flowing material, the smallest flowing units of matter may be ‘molecules’, ‘atoms’, ‘ions’, or ‘electrons’. All of them are quantum particles, or quantons. In short, the quantum of action implies that matter is made of quantons. Matter quantons share some properties with ordinary stones, but also differ from them in many ways. A stone has position and momentum, mass and acceleration, size, shape, structure, orientation and angular mo- mentum, and colour. We now explore each of these properties for quantons, and see how copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net they are related to the quantum of action. Mat ter quantons and their motion – mat ter waves Ref. 40 In 1923 and 1924, the influential physicist Louis de Broglie pondered the consequences of the quantum of action for matter particles. He knew that in the case of light, the quantum of action connects wave behaviour to particle behaviour. He reasoned that the same should apply to matter. It dawned to him that streams of matter particles with the same momentum should behave as waves, just as streams of light quanta do. He thus pre- dicted that like for light, coherent matter flows should have a wavelength 𝜆 and angular frequency 𝜔 given by 2π ℏ 𝐸 𝜆= and 𝜔 = , (21) 𝑝 ℏ where 𝑝 and 𝐸 are the momentum and the energy, respectively, of the single particles. Equivalently, we can write the relations as 𝑝 = ℏ𝑘 and 𝐸 = ℏ𝜔 . (22) All these relations state that matter quantons also behave as waves. For everyday objects, the predicted wavelength is unmeasurably small – though not for microscopic particles. Soon after de Broglie’s prediction, experiments began to confirm it. Matter streams 3 motion of matter – beyond classical physics 77 Motion Mountain – The Adventure of Physics F I G U R E 39 Formation over time of the interference pattern of electrons, here in a low-intensity double-slit experiment: (a) 8 electrons, (b) 270 electrons, (c) 2000 electrons, (d) 6000 electrons, after 20 minutes of exposure. The last image corresponds to the situation shown in the previous ﬁgure. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net (© Tonomura Akira/Hitachi). were observed to diffract, refract and interfere; and all observations matched the values predicted by de Broglie. Because of the smallness of the wavelength of quantons, careful experiments are needed to detect these effects. But one by one, all experimental confirm- ations of the wave properties of light were repeated for matter beams. For example, just as light is diffracted when it passes around an edge or through a slit, matter is also diffrac- ted in these situations. This is true even for electrons, the simplest particles of everyday Ref. 41 matter, as shown in Figure 38. In fact, the experiment with electrons is quite difficult. It was first performed by Claus Jönsson in Tübingen in 1961; in the year 2002 it was voted the most beautiful experiment in all of physics. Many years after Jönsson, the experiment was repeated with a modified electron microscope, as shown in Figure 39. Inspired by light interferometers, researchers began to build matter interferometers. Matter interferometers have been used in many beautiful experiments, as we will find Vol. V, page 142 out. Today, matter interferometers work with beams of electrons, nucleons, nuclei, atoms, Ref. 42 or even large molecules. Just as observations of light interference prove the wave char- Vol. III, page 101 acter of light, so the interference patterns observed with matter beams prove the wave character of matter. They also confirm the value of ℏ. Like light, matter is made of particles; like light, matter behaves as a wave when large numbers of particles with the same momentum are involved. But although beams of large 78 3 motion of matter – beyond classical physics molecules behave as waves, everyday objects – such as cars on a motorway – do not. There are several reasons for this. First, for cars on a motorway the relevant wavelength is extremely small. Secondly, the speeds of the cars vary too much. Thirdly, cars can be counted. In summary, streams of cars with the same speed cannot be made coherent. If matter behaves like a wave, we can draw a strange conclusion. For any wave, the position and the wavelength cannot both be sharply defined simultaneously: the inde- terminacies of the wave number 𝑘 = 2π/𝜆 and of the position 𝑋 obey the relation 1 Δ𝑘Δ𝑋 ≥ . (23) 2 Similarly, for every wave the angular frequency 𝜔 = 2π𝑓 and the instant 𝑇 of its peak amplitude cannot both be sharply defined. Their indeterminacies are related by 1 Δ𝜔Δ𝑇 ≥ . (24) Motion Mountain – The Adventure of Physics 2 Using de Broglie’s wave properties of matter (22), we get ℏ ℏ Δ𝑝Δ𝑋 ⩾ and Δ𝐸Δ𝑇 ⩾ . (25) 2 2 These famous relations are called Heisenberg’s indeterminacy relations. They were dis- covered by Werner Heisenberg in 1925. They are valid for all quantum particles, be they matter or radiation. The indeterminacy relations state that there is no way to simultan- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net eously ascribe a precise momentum and position to a quantum system, nor to simultan- eously ascribe a precise energy and age. The more accurately one quantity is known, the less accurately the other is.* As a result, matter quantons – rather like stones – can always be localized, but always only approximately. On the other hand, we saw that photons of- ten cannot be localized. Both indeterminacy relations have been checked experimentally in great detail. All experiments confirm them. In fact, every experiment proving that matter behaves like a wave is a confirmation of the indeterminacy relation – and vice versa. When two variables are linked by indeterminacy relations, one says that they are com- plementary to each other. Niels Bohr systematically explored all possible such pairs. You Challenge 61 s can also do that for yourself. Bohr was deeply fascinated by the existence of a com- plementarity principle, and he later extended it in philosophical directions. In a well- known scene, somebody asked him what was the quantity complementary to precision. He answered: ‘clarity’. We remark that the usual, real, matter quantons always move more slowly than light. Due to the inherent fuzziness of quantum motion, it should not come to a surprise that exceptions exist. Indeed, in some extremely special cases, the quantum of action allows the existence of particles that move faster than light – so-called virtual particles – which Page 193 we will meet later on. * A policeman stopped the car being driven by Werner Heisenberg. ‘Do you know how fast you were driv- ing?’ ‘No, but I know exactly where I was!’ 3 motion of matter – beyond classical physics 79 In summary, the quantum of action means that matter quantons do not behave like point-like stones, but as waves. In particular, like for waves, the values of position and momentum cannot both be exactly defined for quantons. The values are fuzzy – position and momentum are undetermined. The more precisely one of the two is known, the less precisely the other is known. Mass and acceleration of quantons Matter quantons, like stones, have mass. Indeed, hits by single electrons, atoms or mo- lecules can be detected, if sensitive measurement set-ups are used. Quantons can also be slowed down or accelerated. We have already explored some of these experiments in Vol. III, page 30 the section on electrodynamics. However, quantons differ from pebbles. Using the time– Challenge 62 s energy indeterminacy relation, you can deduce that 2𝑚𝑐3 𝑎⩽ . (26) Motion Mountain – The Adventure of Physics ℏ Thus there is a maximum acceleration for quantons.* Indeed, no particle has ever been Ref. 43 observed with a higher acceleration than this value. In fact, no particle has ever been observed with an acceleration anywhere near this value. The quantum of action thus prevents rest but also limits acceleration. Why are atoms not flat? Why d o shapes exist? The quantum of action determines all sizes in nature. In particular, it determines all copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net shapes. Let us start to explore this topic. Experiments show that all composed quantons, such as atoms or molecules, have structures of finite size and often with complex shape. The size and the shape of every composed quanton are due to the motion of their constituents. The motion of the con- stituents is due to the quantum of action; but how do they move? In 1901, Jean Perrin and independently, in 1904, Nagaoka Hantaro proposed that Ref. 44 atoms are small ‘solar systems’. In 1913, Niels Bohr used this idea, combining it with the quantum of action, and found that he could predict the size and the colour of hy- Ref. 45 drogen atoms, two properties that had not until then been understood. We will perform Page 181 the calculations below. Even Bohr knew that the calculations were not completely un- derstood, because they seemed to assume that hydrogen atoms were flat, like the solar system is. But first of all, atoms are observed to be spherical. Secondly, a flat shape would Challenge 64 e contradict the quantum of action. Indeed, the quantum of action implies that the mo- tion of quantum constituents is fuzzy. Therefore, all composed quantons, such as atoms or molecules, must be made of clouds of constituents. * We note that this acceleration limit is different from the acceleration limit due to general relativity: 𝑐4 𝑎⩽ . (27) 4𝐺𝑚 In particular, the quantum limit (26) applies to microscopic particles, whereas the general-relativistic limit applies to macroscopic systems. Can you confirm that in each domain the relevant limit is the smaller of Challenge 63 e the two? 80 3 motion of matter – beyond classical physics F I G U R E 40 Probability clouds: a hydrogen atom in its spherical ground state (left) and in a non-spherical excited state (right) as seen by an observer travelling around it (QuickTime ﬁlm produced Motion Mountain – The Adventure of Physics with Dean Dauger’s software package ‘Atom in a Box’, available at daugerresearch.com). In short, the quantum of action predicts: ⊳ Atoms are spherical clouds. Experiment and theory confirm that the shape of any atom is due to the cloud, or prob- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ability distribution, of its lightest components, the electrons. The quantum of action thus states that atoms or molecules are not hard balls, as Democritus or Dalton believed, but that they are clouds. Matter is made of clouds. Atomic electron clouds are not infinitely hard, but can to a certain degree interpen- etrate and be deformed. The region where this deformation occurs is called a chemical bond. Bonds lead to molecules. Molecules, being composed of atoms, are composed of (deformed) spherical clouds. Bonds also lead to liquids, solids, flowers and people. A de- tailed exploration confirms that all shapes, from the simplest molecules to the shape of people, are due to the interactions between electrons and nuclei of the constituent atoms. Nowadays, molecular shapes can be calculated to high precision. Small molecules, like water, have shapes that are fairly rigid, though endowed with a certain degree of elasti- city. Large molecules, such as polymers or peptides, have flexible shapes. These shape changes are essential for their effects inside cells and thus for our survival. A large body of biophysical and biochemical research is exploring molecular shape effects. In summary, the quantum of action implies that shapes exist – and that they fluctuate. For example, if a long molecule is held fixed at its two ends, it cannot remain at rest in between. Such experiments are easy to perform nowadays, for example with DNA. In fact, all experiments confirm that the quantum of action prevents rest, produces sizes and shapes, and enables chemistry and life. In nature, all sizes and shapes are due to the quantum of action. Now, every macro- scopic object and every quantum object with a non-spherical shape is able to rotate. We therefore explore what the quantum of action can say about rotation. 3 motion of matter – beyond classical physics 81 𝜃 𝑅 source 𝑎 F I G U R E 41 The quantization of Motion Mountain – The Adventure of Physics angular momentum. Rotation, quantization of angular momentum, and the lack of north poles “ Tristo è quel discepolo che non avanza il suo ” maestro. Leonardo da Vinci* In everyday life, rotation is a frequent type of motion. Wheels are all around us. It turns copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net out that the quantum of action has important consequences for rotational motion. First of all, we note that action and angular momentum have the same physical dimension: both are measured in Js or Nms. It only takes a little thought to show that if matter or radiation has a momentum and wavelength related by the quantum of action, then an- gular momentum is fixed in multiples of the quantum of action. This beautiful argument Ref. 46 is due to Dicke and Wittke. Imagine a circular fence, made of 𝑁 vertical steel bars spaced apart at a distance 𝑎 = 2π𝑅/𝑁, as shown in Figure 41. At the centre of the fence, imagine a source of matter or radiation that can emit particles towards the fence in any chosen direction. The linear momentum of such a particle is 𝑝 = ℏ𝑘 = 2πℏ/𝜆. At the fence slits, the wave will interfere. Outside the fence, the direction of the motion of the particle is determined by the condi- tion of constructive interference. In other words, the angle 𝜃, describing the direction of motion outside the fence, is given by 𝑎 sin 𝜃 = 𝑀𝜆, where 𝑀 is an integer. Through the deflection due to the interference process, the fence receives a linear momentum 𝑝 sin 𝜃, Challenge 65 e or an angular momentum 𝐿 = 𝑝𝑅 sin 𝜃. Combining all these expressions, we find that the angular momentum transferred to the fence is 𝐿 = 𝑁𝑀ℏ . (28) * ‘Sad is that disciple who does not surpass his master.’ This statement from one of his notebooks, the Codice Forster III, is sculpted in large letters in the chemistry aula of the University of Rome La Sapienza. 82 3 motion of matter – beyond classical physics In other words, the angular momentum of the fence is an integer multiple of ℏ. Fences can only have integer intrinsic angular momenta (in units of ℏ). The generalization of the argument to all bodies is also correct. (Of course, this latter statement is only a hint, not a proof.) ⊳ The measured intrinsic angular momentum of bodies is always a multiple of ℏ. Quantum theory thus states that every object’s angular momentum increases in steps. Angular momentum is quantized. This result is confirmed by all experiments. But rotation has more interesting aspects. Thanks to the quantum of action, just as linear momentum is usually fuzzy, so is angular momentum. There is an indeterminacy Ref. 47 relation for angular momentum 𝐿. The complementary variable is the phase angle 𝜑 of Ref. 48 the rotation. The indeterminacy relation can be expressed in several ways. The simplest Page 48 approximation – and thus not the exact expression – is Motion Mountain – The Adventure of Physics ℏ Δ𝐿 Δ𝜑 ⩾ . (29) 2 This is obviously an approximation: the relation is only valid for large angular momenta. In any case, the expression tells us that rotation behaves similarly to translation. The ex- pression cannot be valid for small angular momentum values, as Δ𝜑 by definition cannot grow beyond 2π. In particular, angular-momentum eigenstates have Δ𝐿 = 0.* The indeterminacy of angular momentum appears for all macroscopic bodies. We can say that the indeterminacy appears for all cases when the angular phase of the system can copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net be measured. The quantization and indeterminacy of angular momentum have important con- sequences. Classically speaking, the poles of the Earth are the places that do not move when observed by a non-rotating observer. Therefore, at those places matter would have a defined position and a defined momentum. However, the quantum of action forbids this. There cannot be a North Pole on Earth. More precisely, the idea of a fixed rota- tional axis is an approximation, not valid in general. This applies in particular to rotating quantum particles. Rotation of quantons The effects of the quantum of action on the rotation of microscopic particles, such as atoms, molecules or nuclei, are especially interesting. We note again that action and an- gular momentum have the same units. The precision with which angular momentum * An exact formulation of the indeterminacy relation for angular momentum is ℏ Δ𝐿 Δ𝜑 ⩾ |1 − 2π𝑃(π)| , (30) 2 where 𝑃(π) is the normalized probability that the angular position has the value π. For an angular- momentum eigenstate, one has Δ𝜑 = π/√3 and 𝑃(π) = 1/2π. This exact expression has been tested and Ref. 49 confirmed by experiments. 3 motion of matter – beyond classical physics 83 can be measured depends on the precision of the rotation angle. But if a microscopic particle rotates, this rotation might be unobservable: a situation in fundamental contrast with the case of macroscopic objects. Experiments indeed confirm that many microscopic particles have unobservable rotation angles. For example, in many (but not all) cases, an atomic nucleus rotated by half a turn cannot be distinguished from the unrotated nuc- leus. If a microscopic particle has a smallest unobservable rotation angle, the quantum of action implies that the angular momentum of that particle cannot be zero. It must always be rotating. Therefore we need to check, for each particle, what its smallest unobservable angle of rotation is. Physicists have checked all particles in nature in experiments, and found smallest unobservable angles (depending on the particle type) of 0, 4π, 2π, 4π/3, π, 4π/5, 2π/3 etc. Let us take an example. Certain nuclei have a smallest unobservable rotation angle of half a turn. This is the case for a prolate nucleus (one that looks like a rugby ball) turning around its short axis, such as a 23 Na nucleus. In this case, both the largest observable Motion Mountain – The Adventure of Physics rotation angle and the indeterminacy are thus a quarter turn. Since the change, or action, produced by a rotation is the number of turns multiplied by the angular momentum, we find that the angular momentum of this nucleus is 2 ⋅ ℏ. As a general result, we deduce from the minimum angle values that the angular mo- mentum of a microscopic particle can be 0, ℏ/2, ℏ, 3ℏ/2, 2ℏ, 5ℏ/2, 3ℏ etc. In other words, the intrinsic angular momentum of a particle, usually called its spin, is an integer multiple of ℏ/2. Spin describes how a particle behaves under rotations. How can a particle rotate? At this point, we do not yet know how to picture the rota- tion. But we can feel it – just as we showed that light is made of rotating entities: all matter, copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net including electrons, can be polarized. This is shown clearly by the famous Stern–Gerlach experiment. Silver, Stern and Gerlach – polarization of quantons After a year of hard work, in 1922, Otto Stern and Walther Gerlach* completed a beau- tiful experiment to investigate the polarization of matter quantons. They knew that in- homogeneous magnetic fields act as polarizers for rotating charges. Rotating charges are present in every atom. Therefore they let a beam of silver atoms, extracted from an oven by evaporation, pass an inhomogeneous magnetic field. They found that the beam splits Ref. 50 into two separate beams, as shown in Figure 42. No atoms leave the magnetic field re- gion in intermediate directions. This is in full contrast to what would be expected from classical physics. The splitting into two beams is an intrinsic property of silver atoms; today we know that it is due to their spin. Silver atoms have spin ℏ/2, and depending on their orient- ation in space, they are deflected either in the direction of the field inhomogeneity or against it. The splitting of the beam is a pure quantum effect: there are no intermediate options. Indeed, the Stern–Gerlach experiment provides one of the clearest demonstra- tions that classical physics does not work well in the microscopic domain. In 1922, the * Otto Stern (1888–1969) and Walther Gerlach (1889–1979) worked together at the University of Frankfurt. For his subsequent measurement of the anomalous magnetic moment of the proton, Stern received the Nobel Prize in Physics in 1943, after he had to flee National Socialism. 84 3 motion of matter – beyond classical physics observation classical prediction silver 𝑧 beam N ∂𝐵 ∂𝑧 S aperture Motion Mountain – The Adventure of Physics silver oven beam F I G U R E 42 The Stern–Gerlach experiment. result seemed so strange that it was studied in great detail all over the world. When one of the two beams – say the ‘up’ beam – is passed through a second set-up, copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net all the atoms end up in the ‘up’ beam. The other possible exit, the ‘down’ beam, remains unused in this case. In other words, the up and down beams, in contrast to the original beam, cannot be split further. This is not surprising. But if the second set-up is rotated by π/2 with respect to the first, again two beams – ‘right’ and ‘left’ – are formed, and it does not matter whether the incoming beam is directly from the oven or from the ‘up’ part of the beam. A partially-rotated set-up yields a partial, uneven split. The proportions of the two final beams depend on the angle of rotation of the second set-up. We note directly that if we split the beam from the oven first vertically and then hori- zontally, we get a different result from splitting the beam in the opposite order. (You can Challenge 66 e check this yourself.) Splitting processes do not commute. When the order of two oper- ations makes a difference to the net result, physicists call them non-commutative. Since all measurements are also physical processes, we deduce that, in general, measurements and processes in quantum systems are non-commutative. Beam splitting is direction-dependent. Matter beams behave almost in the same way as polarized light beams. Indeed, the inhomogeneous magnetic field acts on matter somewhat like a polarizer acts on light. The up and down beams, taken together, define a polarization direction. Indeed, the polarization direction can be rotated, with the help of a homogeneous magnetic field. And a rotated beam in a unrotated magnet behaves like an unrotated beam in a rotated magnet. In summary, matter quantons can be polarized. We can picture polarization as the orientation of an internal rotation axis of the massive quanton. To be consistent, the 3 motion of matter – beyond classical physics 85 F I G U R E 43 An idealized graph of the heat capacity of hydrogen over temperature (© Peter Eyland). rotation axis must be imagined to precess around the direction of polarization. Thus, Motion Mountain – The Adventure of Physics massive quantum particles resemble photons also in their polarizability. Curiosities and fun challenges ab ou t quantum mat ter “ It is possible to walk while reading, but not to ” read while walking. Serge Pahaut The quantum of action implies that there are no fractals in nature. Everything is made of particles. And particles are clouds. Quantum theory requires that all shapes in nature be copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ‘fuzzy’ clouds. ∗∗ Can atoms rotate? Can an atom that falls on the floor roll under the table? Can atoms be put into high-speed rotation? The answer is ‘no’ to all these questions, because angular Ref. 51 momentum is quantized; moreover, atoms are not solid objects, but clouds. The macro- scopic case of an object turning more and more slowly until it stops does not exist in the microscopic world. The quantum of action does not allow it. ∗∗ Light is refracted when it enters dense matter. Do matter waves behave similarly? Yes, they do. In 1995, David Pritchard showed this for sodium waves entering a gas of helium Ref. 52 and xenon. ∗∗ Many quantum effects yield curves that show steps. An important example is the molar heat of hydrogen H2 gas, shown in Figure 43. In creasing the temperature from 20 to 8 000 K, the molar heat is shows two steps, first from 3𝑅/2 to 5𝑅/2, and then to 7𝑅/2. Can you explain the reason? ∗∗ Most examples of quantum motion given so far are due to electromagnetic effects. Can 86 3 motion of matter – beyond classical physics you argue that the quantum of action must also apply to nuclear motion, and in particu- Challenge 67 s lar, to the nuclear interactions? ∗∗ There are many other formulations of the indeterminacy principle. An interesting one is due to de Sabbata and Sivaram, who explained in 1992 that the following intriguing relation between temperature and time also holds: Δ𝑇Δ𝑡 ⩾ ℏ/𝑘. (31) Ref. 53 Here, 𝑘 is the Boltzmann constant. All experimental tests so far have confirmed the result. ∗∗ Challenge 68 e Here is a trick question: what is the moment of inertia of an electron? Why? Motion Mountain – The Adventure of Physics First summary on the motion of quantum particles In summary, the ‘digital’ beam splitting seen in the Stern–Gerlach experiment and the wave properties of matter force us to rethink our description of motion. They show that microscopic matter motion follows from the quantum of action, the smallest observ- able action value. In special relativity, the existence of a maximum speed forced us to introduce the concept of space-time, and then to refine our description of motion. In general relativity, the maximum force obliged us to introduce the concepts of horizon and curvature, and then again to refine our description of motion. At the present point, the existence of the quantum of action and the wave behaviour of matter force us to take copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net two similar steps: we first introduce the concept of a wave function, and then we refine our description of matter motion. Chapter 4 T H E QUA N T UM DE S C R I P T ION OF M AT T E R A N D I T S MOT ION “ Die Quanten sind doch eine hoffnungslose Ref. 54 Schweinerei!** Max Born I n everyday life and in classical physics, we say that a system has a position, that Motion Mountain – The Adventure of Physics t is oriented in a certain direction, that it has an axis of rotation, and that t is in a state with specific momentum. In classical physics, we can talk in this way because the state – the situation a system ‘is’ in and the properties a system ‘has’ – coincide with the results of measurement. They coincide because measurements can always be imagined to have a negligible effect on the system. However, because of the existence of a smallest action, the interaction necessary to perform a measurement on a system cannot be made arbitrarily small. Therefore, the quantum of action makes it impossible for us to continue saying that a system has mo- mentum, has position or has an axis of rotation. The quantum of action forces us to use the idea of the rotating arrow and to introduce the concept of wave function or state copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net function. Let us see why and how. States and measurements – the wave function Page 83 The Stern–Gerlach experiment shows that the measured values of spin orientation are not intrinsic, but result from the measurement process (in this case, from the interaction with the applied inhomogeneous field). This is in contrast to the spin magnitude, which is intrinsic and independent of state and measurement. In short, the quantum of action forces us to distinguish carefully three concepts: — the state of the system; — the operation of measurement; — the result or outcome of the measurement. In contrast to the classical, everyday case, the state of a quantum system – the proper- ties a system ‘has’ – is not described by the outcomes of measurements. The simplest illustration of this difference is the system made of a single particle in the Stern–Gerlach experiment. The experiment shows that a spin measurement on a general (oven) particle state sometimes gives ‘up’ (say +1), and sometimes gives ‘down’ (say −1). So a general atom, in an oven state, has no intrinsic orientation. Only after the measurement, an atom is either in an ‘up’ state or in a ‘down’ state. ** ‘Those quanta are a hopeless dirty mess!’ 88 4 the quantum description of matter It is also found that feeding ‘up’ states into a second measurement apparatus gives only ‘up’ states: thus certain special states, called eigenstates, do remain unaffected by measurement. Finally, the Stern–Gerlach experiment and its variations show that states can be ro- tated by applied fields: atom states have a direction or orientation in space. The experi- ments also show that the states rotate as the atoms move through space. The experimental observations can be described in a straightforward way. Since meas- urements are operations that take a state as input and produce an output state and a measurement result, we can say: ⊳ States are described by rotating arrows, or rotating vectors. ⊳ Measurements of observables are operations on the state vectors. ⊳ Measurement results are real numbers; and like in classical physics, they usu- ally depend on the observer. Motion Mountain – The Adventure of Physics In particular, we have distinguished two quantities that are not distinguished in classical physics: states and measurement results. Given this distinction, quantum theory follows quite simply, as we shall see. Given that the quantum of action is not vanishingly small, any measurement of an observable quantity is an interaction with a system and thus a transformation of its state. Therefore, quantum physics describes physical observables as operators, or equivalently, as transformations. The Stern–Gerlach experiment shows this clearly: the interaction with the field influences the atoms: some in one way, and some in another way. In fact, all experiments show: copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ⊳ Mathematically, states are complex vectors, or rotating arrows, in an abstract Page 237 space. This space of all possible states or arrows is a Hilbert space. ⊳ Mathematically, measurements are linear transformations, more precisely, they are described by self-adjoint, or Hermitean, operators (or matrices). ⊳ Mathematically, changes of viewpoint are described by unitary operators (or matrices) that act on states, or arrows, and on measurement operators. Quantum-mechanical experiments also show that a measurement of an observable can only give a result that is an eigenvalue of the corresponding transformation. The result- ing states after the measurement, those exceptional states that are not influenced when the corresponding variable is measured, are the eigenvectors. In short, every expert on motion must know what an eigenvalue and an eigenvector is. For any linear transformation 𝑇, those special vectors 𝜓 that are transformed into multiples of themselves, 𝑇𝜓 = 𝜆𝜓 (32) are called eigenvectors (or eigenstates), and the multiplication factor 𝜆 is called the asso- ciated eigenvalue. Experiments show: ⊳ The state of the system after a measurement is given by the eigenvector cor- responding to the measured eigenvalue. 4 the quantum description of matter 89 In the Stern–Gerlach experiment, the eigenstates are the ‘up’ and the ‘down’ states. In general, the eigenstates are those states that do not change when the corresponding vari- able is measured. Eigenvalues of Hermitean operators are always real, so that consistency is ensured: all measurement results are real numbers. In summary, the quantum of action obliges us to distinguish between three concepts that are mixed together in classical physics: the state of a system, a measurement on the system, and the measurement result. The quantum of action forces us to change the vocabulary with which we describe nature, and obliges to use more differentiated con- cepts. Now follows the main step: the description of motion with these concepts. This is what is usually called ‘quantum theory’. Visualizing the wave function: rotating arrows and probability clouds We just described the state of a quanton with an arrow. In fact, this is an approximation for localized quantons. More precisely, Motion Mountain – The Adventure of Physics ⊳ The state of a quantum particle is described by a spatial distribution of ar- rows, a so-called wave function. To develop a visual image of the wave function, we first imagine a quantum particle that is localized as much as possible. In this case, the wave function for a free quanton can be described simply by a single rotating arrow. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Experiments show that when a localized quanton travels through space, the attached arrow rotates. If the particle is non-relativistic and if spin can be neglected, the rotation takes place in a plane perpendicular to the direction of motion. The end of the arrow then traces a helix around the direction of motion. In this case, the state at a given time is described by the angle of the arrow. This angle is the quantum phase. The quantum phase is responsible for the wave properties of matter, as we will see. The wavelength and the frequency of the helix are determined by the momentum and the kinetic energy of the particle. If the particle is not localized – but still non-relativistic and still with negligible spin effects – the state, or the wave function, defines a rotating arrow at each point in space. The rotation still takes place in a plane perpendicular to the direction of motion. But now we have a distribution of arrows that all trace helices parallel to the direction of motion. At each point in space and time, the state has a quantum phase and a length of the arrow. The arrow lengths decrease towards spatial infinity. Figure 44 shows an example of evolution of a wave function for non-relativistic particles with negligible spin effects. The direction of the arrow at each point is shown by the colour at the specific point. The length of the arrow is shown by the brightness of the colour. For non-relativistic particles with negligible spin effects, the wave function 𝜓(𝑡, 𝑥) is thus described by a length and a phase: it is a complex number at each point in Page 225 space. The phase is essential for interference and many other wave effects. What meas- urable property does the amplitude, the length of the local arrow, describe? The answer was given by the famous physicist Max Born: 90 4 the quantum description of matter F I G U R E 44 The motion of a wave function, the quantum state, through a double slit, showing both the particle and the wave properties of matter. The density of the state, related to the arrow length, is displayed by brightness, and Motion Mountain – The Adventure of Physics the local phase is encoded in the colour. (QuickTime ﬁlm © Bernd Thaller) ⊳ The amplitude of the wave function is a probability amplitude. The square of the amplitude, i.e., the quantity |𝜓(𝑡, 𝑥)|2 , gives the probability to find the particle at the place 𝑥 at time 𝑡. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net In other terms, a wave function is a combination of two ideas: on the one hand, a wave function is a cloud; on the other hand, at each point of the cloud one has to imagine an arrow. Over time, the arrows rotate and the cloud changes shape. ⊳ A wave function is a cloud of rotating arrows. Describing the state of a matter particle with a cloud of rotating arrows is the essential step to picture the wave properties of matter. We can clarify the situation further. ⊳ In every process in which the phase of the wave function is not important, the cloud image of the wave function is sufficient and correct. For example, the motion of atoms of molecules in gases or liquids can be imagined as the motion of cloudy objects. It needs to be stressed that the clouds in question are quite hard: it takes a lot of energy to deform atomic clouds. The hardness of a typical crystal is directly related to the hardness of the atomic clouds that are found inside. Atoms are extremely stiff, or hard clouds. On the other hand, ⊳ In every process in which the phase of the wave function does play a role, 4 the quantum description of matter 91 the cloud image of the wave function needs to be expanded with rotating arrows at each point. This is the case for interference processes of quantons, but also for the precise description of chemical bonds. Teachers often discuss the best way to explain wave functions. Some teachers prefer to use the cloud model only, others prefer not to use any visualization at all. Both ap- proaches are possible; but the most useful and helpful approach is to imagine the state or wave function of a non-relativistic quantum particle as an arrow at every point in space. The rotation frequency of the set of arrows is the kinetic energy of the particle; the wavelength of the arrow motion – the period of the helical curve that the tip of the ar- rows – or of the average arrow – traces during motion – is the momentum of the quantum particle. An arrow at each point in space is a (mathematical) field. The field is concentrated in the region where the particle is located, and the amplitude of the field is related to Motion Mountain – The Adventure of Physics the probability to find the particle. Therefore the state field, the wave function or state function, is an arrow cloud. It is usually called with the greek letter 𝜓. Note that even though the wave function can be seen as defining an arrow at every point in space, the wave function as a whole can also be described as one, single vec- Page 237 tor, this time in a Hilbert space. For free particles, i.e., particles that are not subject to external forces, the Hilbert space is infinite dimensional! Nevertheless, it is not hard to calculate in such spaces. The scalar product of two wave functions is the spatial integ- ral of the product of the complex conjugate of the first function and the (unconjugated) second function. With this definition, all vector concepts (unit vectors, null vectors, basis copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Challenge 69 e vectors, etc.) can be meaningfully applied to wave functions. In summary, for non-relativistic particles without spin effects, the state or wave func- tion of a quantum particle is a cloud, or a distributed wave, of rotating arrows. This aspect of a quantum cloud is unusual. Since a quantum cloud is made of little arrows, every point of the cloud is described by a local density and a local orientation. This latter prop- erty does not occur in any cloud of everyday life. For many decades it was tacitly assumed that a wave function 𝜓 cannot be visual- ized more simply than with a cloud of rotating arrows. Only the last years have shown that there are other visualization for such quantum clouds; one possible visualization is Vol. VI, page 174 presented in the last volume of this series. The state evolu tion – the S chrödinger equation The description of the state of a non-relativistic quanton with negligible spin effects as a rotating cloud completely determines how the wave function evolves in time. Indeed, for such quantum particles the evolution follows from the total energy, the sum of kinetic and potential energy 𝑇 + 𝑉, and the properties of matter waves: ⊳ The local rate of change of the state arrow(s) 𝜓(𝑥), or simply 𝜓, is produced 92 4 the quantum description of matter by the local total energy, or Hamiltonian, 𝐻 = 𝑇 + 𝑉: ∂ 𝑖ℏ 𝜓 = 𝐻𝜓 . (33) ∂𝑡 This famous equation is Schrödinger’s equation of motion.* This evolution equation ap- plies to all quantum systems and is one of the high points of modern physics. Ref. 55 In fact, Erwin Schrödinger had found his equation in two different ways. In his first Ref. 56 paper, he deduced it from a variational principle. In his second paper, he deduced the evolution equation directly, by asking a simple question: how does the state evolve? He knew that the state of a quanton behaves both like a wave and like a particle. A wave is described by a field, which he denoted 𝜓(𝑡, 𝑥). If the state 𝜓 behaves like a wave, then the corresponding wave function must be an amplitude 𝑊 multiplied by a phase factor e𝑖𝑘𝑥−𝜔𝑡 . The state can thus be written as Motion Mountain – The Adventure of Physics 𝜓(𝑡, 𝑥) = 𝑊(𝑡, 𝑥) e𝑖𝑘𝑥−𝜔𝑡 . (34) The amplitude 𝑊 is the length of the local arrow; the phase is the orientation of the local arrow. Equivalently, the amplitude is the local density of the cloud, and the phase is the local orientation of the cloud. We know that the quantum wave must also behave like a particle of mass 𝑚. In par- ticular, the non-relativistic relation between energy and momentum 𝐸 = 𝑝2 /2𝑚 + 𝑉(𝑥) – where 𝑉(𝑥) is the potential at position 𝑥 – must be fulfilled for these waves. The two Page 76 de Broglie relations (22) for matter wavelength and matter frequency then imply copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ∂𝜓 −ℏ2 2 𝑖ℏ = 𝐻𝜓 = ∇ 𝜓 + 𝑉(𝑥)𝜓 . (35) ∂𝑡 2𝑚 This is the complete form of Schrödinger’s wave equation. ∇2 is the Laplace operator, es- sentially the second derivative over space. It states how the arrow wave, the wave function 𝜓 associated to a particle, evolves over time. In 1926, this wave equation for the complex field 𝜓 became instantly famous when Schrödinger used it, by inserting the potential felt by an electron near a proton, to calculate the energy levels of the hydrogen atom. In a hy- drogen atom, light is emitted by the single electron inside that atom; therefore a precise description of the motion of the electron in a hydrogen atom allows us to describe the light frequencies it can emit. (We will perform the calculation and the comparison with Page 181 experiment below.) First of all, the Schrödinger equation explained that only discrete col- ours are emitted by hydrogen. In addition, the frequencies of the emitted light were found to be in agreement with the prediction of the equation to five decimal places. Finally, the * Erwin Schrödinger (b. 1887 Vienna, d. 1961 Vienna) was famous for being a physicien bohémien, always living in a household with two women. In 1925 he discovered the equation that brought him international fame, and the Nobel Prize in Physics in 1933. He was also the first to show that the radiation discovered by Victor Hess in Vienna was indeed coming from the cosmos. He left Germany, and then again Austria, out of dislike for National Socialism, and was a professor in Dublin for many years. There he published his famous and influential book What is life?. In it, he came close to predicting the then-unknown nucleic acid DNA from theoretical insight alone. 4 the quantum description of matter 93 F I G U R E 45 Erwin Schrödinger (1887 –1961) Motion Mountain – The Adventure of Physics size of atoms was predicted correctly. These were important results, especially if we keep in mind that classical physics cannot even explain the existence of atoms, let alone their light emission! In contrast, quantum physics explains all properties of atoms and their colours to high precision. In other words, the discovery of the quantum of action led the description of the motion of matter to a new high point. In fact, the exact description of matter quantons is only found when both spin ef- fects and the relativistic energy–momentum relation are taken into account. We do this Page 188 below. No deviations between the full relativistic calculations and experiments have ever copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net been found. And even today, predictions and measurements of atomic spectra remain the most precise and accurate in the whole study of nature: in the cases that experimental precision allows it, the calculated values agree with experiments to 13 decimal places. Self-interference of quantons Page 76 Waves interfere. All experiments, including the examples shown in Figure 38 and Fig- ure 39, confirm that all quantum particles, and in particular all matter quantons, show interference. Interference is a direct consequence of the Schrödinger equation, as the Page 90 film of Figure 44 shows. The film illustrates the solution of the Schrödinger equation for a quantum particle moving through a double slit. The film visualizes how a double slit induces diffraction and interference for a matter particle. It turns out that the Schrödinger equation completely reproduces and explains the observations of matter interference: also the interference of matter quantons is due to the evolution of clouds of rotating arrows. And like in all interference phenomena, the local intensity of the interference pattern turns out to be proportional to the square |𝑊|2 of the local wave amplitude. And the local wave amplitude results from the phase of the interfering wave trains. The analogy with light interefence is complete; even the formulae are the same. We note that even though the wave function is spread out over the whole detection screen just before it hits the screen, it nevertheless yields only a localized spot on the Page 153 screen. This effect, the so-called collapse of the wave function, is explored in detail below. 94 4 the quantum description of matter F I G U R E 46 The evolution of a wave function (lowest curve) with zero momentum, and the motion of its parts with positive and negative Motion Mountain – The Adventure of Physics momenta. Local phase is encoded in the colour. (QuickTime ﬁlm © Bernd Thaller) The speed of quantons Let us delve a little into the details of the description given by the Schrödinger equation copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net (35). The equation expresses a simple connection: the classical speed of a matter particle is the group velocity of the wave function 𝜓. Seen from far away, the wave function thus moves like a classical particle would. But we know from classical physics that the group velocity is not always well defined: in cases where the group dissolves into several peaks, the concept of group velocity is not of much use. These are also the cases in which quantum motion is very different from Page 153 classical motion, as we will soon discover. But for well-behaved cases, such as free or almost free particles, we find that the wave function moves in the same way as a classical particle does. The Schrödinger equation makes another point: velocity and position of matter are not independent variables, and cannot be chosen at will. The initial condition of a system is given by the initial value of the wave function alone. No derivatives have to be (or can be) specified. Indeed, experiments confirm that quantum systems are described by a first-order evolution equation, in stark contrast to classical systems. The reason for this contrast is the quantum of action and the limit it poses on the possible state variables of a particle. Dispersion of quantons For free quantum particles, the Schrödinger’s evolution equation implies dispersion, as illustrated in Figure 46. Imagine a wave function that is localized around a given starting position. Such a wave function describes a quantum system at rest. When time passes, 4 the quantum description of matter 95 F I G U R E 47 The tunnelling of a wave function through a potential hill (the rectangular column): most of the wave function is reﬂected, and part of the Motion Mountain – The Adventure of Physics wave function passes to the other side. Local phase is encoded in the colour. (QuickTime ﬁlm © Bernd Thaller) this wave function will spread out in space. Indeed, Schrödinger’s evolution equation is similar, mathematically, to a diffusion equation. In the same way that a drop of ink in water spreads out, also the state of a localized quantum particle will spread out in copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net space. True, the most probable position stays unchanged, but the probability to find the particle at large distances from the starting position increases over time. For quantum particles, this spreading effect is indeed observed by all experiments. The spread is a consequence of the wave aspect of matter, and thus of the quantum of action ℏ. It occurs Challenge 70 e for quantons at rest and therefore also for quantons in motion. For macroscopic objects, the spreading effect is not observed, however: cars rarely move away from parking spaces. Indeed, quantum theory predicts that for macroscopic systems, the effect of spreading is Challenge 71 ny negligibly small. Can you show why? In summary, the wave aspect of matter leads to the spreading of wave functions. Wave functions show dispersion. Tunnelling and limits on memory – damping of quantons ‘Common sense’ says that a slow ball cannot roll over a high hill. More precisely, classical physics says that if the kinetic energy 𝑇 is smaller than the potential energy 𝑉 that the ball would have at the top of the hill, then the ball cannot reach the top of the hill. In contrast, according to quantum theory, there is a non-vanishing probability of passing the hill for any energy of the ball. In quantum theory, hills and obstacles are described by potential barriers, and objects by wave functions. Any initial wave function will spread beyond any potential barrier of finite height and width. The wave function will also be non-vanishing at the location of the barrier. In short, any object can overcome any hill or barrier, as shown in Figure 48. 96 4 the quantum description of matter E m p 0 Δx F I G U R E 48 Climbing a hill. This effect is called the tunnelling effect. It is in complete contrast to everyday experience Motion Mountain – The Adventure of Physics – and to classical mechanics. The tunnelling effect results from a new aspect contained in the quantum descrip- tion of hills: in nature, any obstacle can be overcome with a finite effort. No obstacle is infinitely difficult to surmount. Indeed, only for a potential of infinite height would the wave function vanish and fail to spread to the other side. But such potentials exist only as approximations; in nature potentials are always of finite value. Challenge 72 ny How large is the tunnelling effect? Calculation shows that the transmission probability 𝑃 is given approximately by 2𝑤 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 16𝑇(𝑉 − 𝑇) − √2𝑚(𝑉 − 𝑇) 𝑃≈ e ℏ (36) 𝑉2 where 𝑤 is the width of the hill, 𝑉 its height, and 𝑚 and 𝑇 the mass and the kinetic energy of the particle. For a system of large number of particles, the probability is (at most) the product of the probabilities for the different particles. Let us take the case of a car in a garage, and assume that the car is made of 1028 atoms at room temperature. A typical garage wall has a thickness of 0.1 m and a potential height of 𝑉 = 1 keV = 160 aJ for the passage of an atom. We get that the probability of finding the car outside the garage is 12 (1028 ) 40 𝑃 ≈ (10−(10 ) ) ≈ 10−(10 ) . (37) Challenge 73 e The smallness of this value (just try to write it down, to be convinced) is the reason why it is never taken into account by the police when a car is reported missing. (Actually, the probability is even considerably smaller. Can you name at least one effect that has been Challenge 74 s forgotten in this simple calculation?) Obviously, tunnelling can be important only for small systems, made of a few particles, and for thin barriers, with a thickness of the order of ℏ/√2𝑚(𝑉 − 𝑇) . For ex- ample, tunnelling of single atoms is observed in solids at high temperature, but is not important in daily life. For electrons, the effect is more pronounced: the barrier width 𝑤 4 the quantum description of matter 97 Farady cage with high screen with electric intereference potential pattern that depends on potential beam splitter F I G U R E 49 A localized electric potential charged matter beam in an interferometer leads to a shift of the interference pattern. Motion Mountain – The Adventure of Physics for an appreciable tunnelling effect is 0.5 nm √aJ 𝑤≈ . (38) √𝑉 − 𝑇 At room temperature, the kinetic energy 𝑇 is of the order of 6 zJ; increasing the temperat- ure obviously increases the tunnelling. As a result, electrons tunnel quite easily through barriers that are a few atoms in width. Indeed, every TV tube uses tunnelling at high temperature to generate the electron beam producing the picture. The necessary heating is the reason why in the past, television tubes took some time to switch on. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net The tunnelling of electrons also limits the physical size of computer memories. Mem- ory chips cannot be made arbitrary small. Silicon integrated circuits with one terabyte of Challenge 75 s random-access memory (RAM) will probably never exist. Can you imagine why? In fact, tunnelling limits the working of any type of memory, including that of our brain. Indeed, if we were much hotter than 37°C, we could not remember anything! Since light is made of particles, it can also tunnel through potential barriers. The best – or highest – potential barriers for light are mirrors; mirrors have barrier heights of the order of one attojoule. Tunnelling implies that light can be detected behind any mirror. These so-called evanescent waves have indeed been detected; they are used in various high-precision experiments and devices. The quantum phase We have seen that the amplitude of the wave function, the probability amplitude, shows the same effects as any wave: dispersion and damping. We now return to the phase of the wave function and explore it in more detail. Whereas the amplitude of a wave function is easy to picture – just think of the (square root of the) density of a real cloud – the phase takes more effort. As mentioned, states or wave functions are clouds with a local phase: they are clouds of rotating arrows, i.e., clouds of objects that rotate and can be rotated. In case of an everyday water cloud, a local rotation of droplets has no effect on the cloud. In contrast, in quantum theory, the local rotation of the cloud, thus the local change of its phase, does have a measurable 98 4 the quantum description of matter solenoid with current screen with intereference pattern that depends on magnetic field beam splitter neutral matter beam F I G U R E 50 Magnetic ﬁelds change the phase of a spinning particle. Motion Mountain – The Adventure of Physics effect. Let us explore this point. Page 56 The phase of free matter waves behaves like the phase of photons: it evolves with time, and thus increases along the path of a moving particle. The phase can be pictured by a small rotating arrow. The angular velocity with which the phase rotates is given by the famous relation 𝜔 = 𝐸/ℏ. In short, ⊳ We can picture the wave function of a free quantum particle as a moving cloud of arrows; the arrows rotate with constant frequency while the cloud disperses at the same time. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Above all, the phase is that aspect of the wave function that leads to interference effects. When two partial wave functions are separated and recombined after a relative phase change, the phase change will determine the interference pattern. This is the origin of the electron beam interference observations shown in Figure 38. Without the quantum phase, there would be no extinction and no interference. The phase of a wave function can be influenced in many ways. The simplest way is the use of electric fields. If the wave function of a charged particle is split, and one part is led through a region with an electric field, a phase change will result. The arrangement is shown in Figure 49. A periodic change of the electric potential should yield a periodic shift of the interference pattern. This is indeed observed. Another simple case of phase manipulation is shown in Figure 50: also a magnetic field changes the phase of a spinning neutral particle – if it contains charges – and thus influences the interference behaviour. A famous experiment shows the importance of the phase in an even more surpris- Ref. 57 ing way: the Aharonov–Bohm effect. The effect is famous for two reasons: it is counter- intuitive and it was predicted before it was observed. Look at the set-up shown in Fig- ure 51. A matter wave of charged particles is split into two by a cylinder – positioned at a right angle to the matter’s path – and the matter wave recombines behind it. Inside the cylinder there is a magnetic field; outside, there is none. (A simple way to realize such a cylinder is a long solenoid.) Quantum physics predicts that an interference pattern will be observed, and that the position of the stripes will depend on the value of the mag- 4 the quantum description of matter 99 magnetic field (even screen if only inside the solenoid) current vector potential charged matter beam Motion Mountain – The Adventure of Physics F I G U R E 51 The Aharonov–Bohm effect: the inﬂuence of the magnetic vector potential on interference (left) and a measurement conﬁrmation (right), using a microscopic sample that transports electrons in thin metal wires (© Doru Cuturela). copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net F I G U R E 52 The motion of a wave function around a solenoid showing the Aharonov–Bohm effect. The density of the state is displayed by brightness, and the local phase is encoded in the colour. (QuickTime ﬁlm © Bernd Thaller) netic field. This happens even though the wave never enters the region with the field! The surprising effect has been observed in countless experiments. The reason for the Aharonov–Bohm effect is simple: for a charged particle, the phase 100 4 the quantum description of matter electrically charged wire screen with intereference pattern that depends on wire charge F I G U R E 53 The Aharonov–Casher effect: beam splitter the inﬂuence of charge on the phase polarized neutron beam leads to interference even for interfering neutrons. Motion Mountain – The Adventure of Physics of a wave function is determined by the vector potential 𝐴, not by the magnetic field 𝐵. The vector potential around a solenoid does not vanish – as we know from the section Vol. III, page 83 on electrodynamics – but circulates around the solenoid. This circulation distinguishes the two sides of the solenoid and leads to a phase shift – one that indeed depends on the magnetic field value – and thus produces interference, even though the particle never interacts with the magnetic field itself. A further example for phase manipulation is the so-called Aharonov–Casher effect, which even occurs for neutral particles, as long as they have a magnetic moment, such as neutrons have. The phase of a polarized neutron will be influenced by an electric field, so copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net that the arrangement shown in Figure 53 will show an interference pattern that depends on the applied electric potential. Another case of phase manipulation will be presented later on: also gravitational fields can be used to rotate wave functions. Even the acceleration due to rotational motion can do so. In fact, it has been possible to measure the rotation of the Earth by observing the Ref. 58 change of neutron beam interference patterns. Another important class of experiments that manipulate the phase of wave functions are possible with macroscopic quantum phenomena. In superconductivity and in super- fluidity, the phase of the wave function is regularly manipulated with magnetic and elec- tric fields. This possibility has many important technical applications. For example, the so-called Josephson effect is used to measure electric potential differences by measuring the frequency of emitted radio waves, and so-called superconducting quantum interfer- ence devices, or SQIDs, are used to measure tiny magnetic fields. We note that all these experiments confirm that the absolute phase of a wave function cannot be measured. However, relative phases – phase differences or phase changes – can Challenge 76 e be measured. Can you confirm this? All the phase shift effects just presented have been observed in numerous experi- ments. The phase is an essential aspect of the wave function: the phase leads to inter- ference and is the main reason for calling it wave function in the first place. Like in any wave, the phase evolves over time and it can be influenced by various external influ- ences. Above all, the experiments show that a localized quantum particle – thus when the spread of the wave function can be neglected – is best imagined as a rotating arrow; 4 the quantum description of matter 101 Motion Mountain – The Adventure of Physics F I G U R E 54 An electron hologram of DNA molecules (© Hans-Werner Fink/Wiley VCH). in contrast, whenever the spread cannot be neglected, the wave function is best imagined as a wave of arrows rotating at each point in space. C an t wo electron beams interfere? Are there coherent electron beams? copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Ref. 59 Do coherent electron sources exist? The question is tricky. Results in the literature, such as the one illustrated in Figure 54, state that is possible to make holograms with electron beams.* However, when one asks these authors about the meaning of coherence, they answer that electron coherence is only transversal, not longitudinal. Transversal coher- ence is determined by the possible size of wavefronts with a given phase. The upper limit of this size is given by the interactions such a state has with its environment. All this behaviour is as expected for actual coherence. However, the concept of ‘transversal coherence’ is a misnomer. The ability to interfere with oneself, as implies in the term ‘transversal coherence’ is not the correct definition of coherence. Transversal coherence, be it for photons or for matter particles, only expresses the smallness of the particle source. Both small lamps (and lasers) can show interference when the beam is split and recombined with identical path length; this is not a proof of coherence of the light field. A similar reasoning shows that monochromaticity is not a proof for coherence either. A state is called coherent if it possesses a well-defined phase throughout a given do- main of space or time. The size of the spatial region or of the time interval defines the degree of coherence. This definition yields coherence lengths of the order of the source size for small ‘incoherent’ sources. Even for a small coherence length, the size of an in- terference pattern or the distance 𝑑 between its maxima can be much larger than the Ref. 60 * In 2002, the first holograms have been produced that made use of neutron beams. 102 4 the quantum description of matter coherence length 𝑙 or the source size 𝑠. In short, a large size (or a persistent duration in time) of an interference pattern alone is not a proof of coherence. Let us recall the situation for light. A light source is coherent if it produces an ap- proximate sine wave over a certain length or time. Due to the indeterminacy relation, in Page 47 any coherent beam of light, the photon number is undetermined. The same requirement applies to coherent electron beams: an undetermined electron number is needed for co- herence. That is impossible, as electrons carry a conserved charge. Coherent electron beams do not exist. In summary, even though an electron can interfere with itself, and even though it is possible to produce interference between two light sources, interference between two electron sources is impossible. Indeed, nobody has every managed to produce interfer- ence between two electron sources. There is no conventional concept of coherence for electron beams. The least action principle in quantum physics Motion Mountain – The Adventure of Physics In nature, motion happens in a way that minimizes change. Indeed, in classical phys- Vol. I, page 253 ics, the principle of least action – or principle of cosmic lazyness – states: in nature, the motion of a particle happens along that particular path – out of all possible paths with the same end points – for which the action is minimal. This principle of cosmic laziness or cosmic efficiency was stated mathematically by saying that in nature, the variation 𝛿𝑆 of the action is zero. Action or change minimization explains all classical evolution equations. We now transfer this idea to the quantum domain. For quantum systems, we need to redefine both the concept of action and the concept of variation: first of all, we have to find a description of action that is based on operators; copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net secondly, we need to define the action variation without paths, as the concept of ‘path’ does not exist for quantum systems; thirdly, since there is a smallest action in nature, a vanishing variation is not a clearly defined concept, and we must overcome this hurdle. There are two main ways to achieve this goal: to describe the motion of quantum sys- tems as a superposition of all possible paths, or to describe action with the help of wave functions. Both approaches are equivalent. In the first approach, the path integral formulation, the motion of a quantum particle is described as a democratic superposition of motions along all possible paths. (We called Page 56 it the ‘arrow model’ above.) For each path, the evolution of the arrow is determined, and at the end point, the arrows from all paths are added. The action for each path is the number of turns that the arrow performs along the path. The result from this exercise is that the path for which the arrow makes the smallest number of turns is usually (but not always!) the most probable path. A more precise investigation shows that classical, macroscopic systems always follow only the path of smallest action, whereas quantum systems follow all paths. In the second approach to quantum physics, action is defined with the help of wave functions. In classical physics, we defined the action (or change) as the integral of the Lagrangian between the initial and final points in time, and the Lagrangian itself as the Vol. I, page 248 difference between kinetic and potential energy. In quantum physics, the simplest defin- ition is the quantum action defined by Julian Schwinger. Let us call the initial and final 4 the quantum description of matter 103 states of the system 𝜓i and 𝜓f . The action 𝑆 between these two states is defined as 𝑆 = ⟨𝜓i | ∫𝐿 d𝑡 | 𝜓f ⟩ , (39) where 𝐿 is the Lagrangian (operator). The angle brackets represent the ‘multiplication’ of states and operators as defined in quantum theory.* In simple words, also in quantum theory, action – i.e., the change occurring in a system – is the integral of the Lagrangian. The Lagrangian operator 𝐿 is defined in the same way as in classical physics: the Lag- rangian 𝐿 = 𝑇−𝑉 is the difference between the kinetic energy 𝑇 and the potential energy 𝑉 operators. The only difference is that, in quantum theory, the momentum and position variables of classical physics are replaced by the corresponding operators of quantum physics.** To transfer the concept of action variation 𝛿𝑆 to the quantum domain, Julian Schwinger introduced the straightforward expression Motion Mountain – The Adventure of Physics 𝛿𝑆 = ⟨𝜓i | 𝛿∫𝐿 d𝑡| 𝜓f ⟩ . (40) The concept of path is not needed in this expression, as the variation of the action is based on varying wave functions instead of varying particle paths. The last classical requirement to be transferred to the quantum domain is that, be- cause nature is lazy, the variation of the action must vanish. However, in the quantum domain, the variation of the action cannot be zero, as the smallest observable action is the quantum of action. As Julian Schwinger discovered, there is only one possible way to copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net express the required minimality of action: 𝛿𝑆 = ⟨𝜓i | 𝛿∫𝐿 d𝑡| 𝜓f ⟩ = −𝑖ℏ 𝛿⟨𝜓i |𝜓f ⟩ . (41) This so-called quantum action principle describes all motion in the quantum domain. Classically, the right-hand side is zero – since ℏ is taken to be zero – and we then re- cover the minimum-action principle 𝛿𝑆 = 0 of classical physics. But in quantum the- ory, whenever we try to achieve small variations, we encounter the quantum of action and changes of (relative) phase. This is expressed by the right-hand side of the expres- sion. The right side is the reason that the evolution equations for the wave function – Schrödinger’s equation for the spinless non-relativistic case, or Dirac’s equation for the spin 1/2 relativistic case – are valid in nature. In other words, all quantum motion – i.e., the quantum evolution of a state 𝜓 or |𝜓⟩ – happens in such a way that the action variation is the same as −𝑖 times the quantum of action ℏ times the variation of the scalar product between initial and final states. In * We skip the details of notation and mathematics here; in the simplest description, states are wave func- tions, operators act on these functions, and the product of two different brackets is the integral of the func- tion product over space. ** More precisely, there is also a condition governing the ordering of operators in a mixed product, so that the non-commutativity of operators is taken into account. We do not explore this issue here. 104 4 the quantum description of matter simple terms, in the actual motion, the intermediate states are fixed by the requirement that they must lead from the initial state to the final state with the smallest number of effective turns of the state phase. The factor −𝑖 expresses the dependence of the action on the rotation of the wave function. In summary, the least action principle is also valid in quantum physics, provided one takes into account that action values below ℏ cannot be found in experiments. The least action principle governs the evolution of wave function. The least action principle thus explains the colour of all things, all other material science, all chemistry and all biology, as we will see in the following. The motion of quantons with spin “ ” Everything turns. Anonymous What is the origin of the quantum phase? Classical physics helps to answer the question. Motion Mountain – The Adventure of Physics Like everyday objects, also quantons can rotate around an axis: we speak of particle spin. Page 82 But if quantum particles can spin, they should possess angular momentum. And indeed, experiments confirm this deduction. In particular, electrons have spin. The full details of electron spin were deduced from Ref. 61 experiments by two Dutch students, George Uhlenbeck and Samuel Goudsmit, in 1925. They had the guts to publish what Ralph Kronig had also suspected: that electrons rotate around an axis with a projected component of the angular momentum given by ℏ/2. In fact, this value – often called spin 1/2 for short – is valid for all elementary matter particles. (In contrast, all known elementary radiation particles have spin values of ℏ, or copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net spin 1 for short.) If a spinning particle has angular momentum, it must be possible to rearrange the axis by applying a torque, to observe precession, to transfer the spin in collisions, etc. All these effects are indeed observed; for example, the Stern–Gerlach experiment already allows Page 83 all these observations. The only difference between particle spin and classical angular Page 82 momentum is that particle spin is quantized, as we deduced above. In other words, the spin 𝐿 of a quantum particle has all the properties of a rotation around an axis. As a consequence, spinning charged quantum particles act as small di- pole magnets, with the magnet oriented along the axis of rotation. The observed strength of the dipole magnet, the magnetic moment, is proportional to the spin and to the con- version factor −𝑒/2𝑚𝑒 , as expected from classical physics. Therefore, the natural unit for the magnetic moment of the electron is the quantity 𝜇B = 𝑒ℏ/2𝑚𝑒 ; it is called Bohr’s mag- neton. It turns out that the magnetic moment 𝜇 of quantons behaves differently from that of classical particles. The quantum effects of spin are described by the so-called 𝑔-factor, which is a pure number: −𝑒 𝐿 𝑒ℏ 𝜇=𝑔 𝐿 = −𝑔𝜇B , with 𝜇B = . (42) 2𝑚𝑒 ℏ 2𝑚𝑒 From the observed optical spectra, Uhlenbeck and Goudsmit deduced a 𝑔-factor of 2 for Page 107 the electron. Classically, one expects a value 𝑔 = 1. The experimental value 𝑔 = 2 was 4 the quantum description of matter 105 Ref. 62 explained by Llewellyn Thomas as a relativistic effect a few months after its experimental discovery. By 2004, experimental techniques had become so sensitive that the magnetic effect of a single electron spin attached to an impurity (in an otherwise non-magnetic ma- terial) could be detected. Researchers now hope to improve these so-called ‘magnetic- resonance-force microscopes’ until they reach atomic resolution. In 1927, Wolfgang Pauli* discovered how to include spin 1/2 in a quantum-mechanical description: instead of a state function described by a single complex number, a state function with two complex components is needed. The reason for this expansion is simple. In general, the little rotating arrow that describes a quantum state does not ro- tate around a fixed axis, as is assumed by the Schrödinger equation; the axis of rotation has also to be specified at each position in space. This implies that two additional para- meters are required at each space point, bringing the total number of parameters to four real numbers, or, equivalently, two complex numbers. Nowadays, Pauli’s equation for quantum mechanics with spin is mainly of conceptual interest, because – like that of Motion Mountain – The Adventure of Physics Schrödinger – it does not comply with special relativity. In summary, the non-relativistic description of a quanton with spin implies the use of wave functions that specify two complex numbers at each point in space and time. The additional complex number describe the local rotation plane of the spin. The idea of including the local rotation plane was also used by Dirac when he introduced the re- lativistic description of the electron, and the idea is also used in all other wave equations for particles with spin. R elativistic wave equations copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net In 1899, Max Planck had discovered the quantum of action. In 1905, Albert Einstein pub- lished the theory of special relativity, which was based on the idea that the speed of light 𝑐 is independent of the speed of the observer. The first question Planck asked himself was whether the value of the quantum of action would be independent of the speed of the observer. It was his interest in this question that led him to invite Einstein to Berlin. With this invitation, he made the patent-office clerk famous in the world of physics. Experiments show that the quantum of action is indeed independent of the speed of the observer. All observers find the same minimum value. To include special relativity into quantum theory, we therefore need to find the correct quantum Hamiltonian 𝐻, i.e., the correct energy operator. * Wolfgang Ernst Pauli (b. 1900 Vienna, d. 1958 Zürich), at the age of 21, wrote one of the best texts on special and general relativity. He was the first to calculate the energy levels of hydrogen using quantum theory, discovered the exclusion principle, incorporated spin into quantum theory, elucidated the relation between spin and statistics, proved the CPT theorem, and predicted the neutrino. He was admired for his intelligence, and feared for his biting criticisms, which led to his nickname, ‘conscience of physics’. Des- pite this trait, he helped many people in their research, such as Heisenberg with quantum theory, without Ref. 63 claiming any credit for himself. He was seen by many, including Einstein, as the greatest and sharpest mind of twentieth-century physics. He was also famous for the ‘Pauli effect’, i.e., his ability to trigger disasters in laboratories, machines and his surroundings by his mere presence. As we will see shortly, one can argue that Pauli actually received the Nobel Prize in Physics in 1945 – officially ‘for the discovery of the exclusion principle’ – for finally settling the question of how many angels can dance on the tip of a pin. 106 4 the quantum description of matter For a free relativistic particle, the classical Hamiltonian function – that is, the energy of the particle – is given by 𝐻 = ±√𝑐4 𝑚2 + 𝑐2 𝑝2 with 𝑝 = 𝛾𝑚𝑣 . (43) Thus we can ask: what is the corresponding Hamilton operator for the quantum world? The simplest answer was given, in 1949 by T.D. Newton and E.P. Wigner, and in 1950, by Ref. 64 L.L. Foldy and S.A. Wouthuysen. The operator is almost the same one: 1 0 0 0 4 2 2 2 0 1 0 0 𝐻 = 𝛽√𝑐 𝑚 + 𝑐 𝑝 with 𝛽 = ( ) . (44) 0 0 −1 0 0 0 0 −1 The signs appearing in the matrix operator 𝛽 distinguish, as we will see, between particles Motion Mountain – The Adventure of Physics and antiparticles. The numbers +1 and −1 appear twice, to take care of the two possible spin directions for each case. With this relativistic Hamiltonian operator for spin 1/2 particles – and with all others – the wave function is described by four complex numbers, two for particles and two for antiparticles. Why? We saw above that a quantum particle with spin requires two Page 105 complex components for its state; this followed from the requirement to specify, at each Vol. II, page 72 point in space, the length of the arrow, its phase, and its plane of rotation. Earlier on we also found that relativity automatically introduces antimatter. (We will explore the issue Page 192 in more detail below.) Both matter and antimatter are thus part of any relativistic de- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net scription of quantum effects. The wave function for a particle has vanishing antiparticle components, and vice versa. In total, the wave function for relativistic spin 1/2 particle has thus four complex components. The Hamilton operator yields the velocity operator 𝑣 through the same relation that is valid in classical physics: d 𝑝 𝑣= 𝑥=𝛽 . (45) d𝑡 √𝑐4 𝑚2 + 𝑐2 𝑝2 This velocity operator shows a continuum of eigenvalues, from minus to plus the speed of light. The velocity 𝑣 is a constant of motion, as are the momentum 𝑝 and the energy 𝐸 = √𝑐4 𝑚2 + 𝑐2 𝑝2 . (46) Also the orbital angular momentum 𝐿 is defined as in classical physics, through 𝐿 =𝑥×𝑝. (47) Ref. 65 The orbital angular momentum 𝐿 and the spin 𝜎 are separate constants of motion. A particle (or antiparticle) with positive (or negative) angular momentum component has 4 the quantum description of matter 107 a wave function with only one non-vanishing component; the other three components vanish. But alas, the representation of relativistic motion named after Foldy and Wouthuysen is not the simplest when it comes to take electromagnetic interactions into account. The simple identity between the classical and quantum-mechanical descriptions is lost when electromagnetism is included. We will solve this problem below, when we explore Dirac’s Page 189 evolution equation for relativistic wave functions. B ound motion, or composite vs. elementary quantons When is an object composite, and not elementary? Whenever it contains internal, or bound motion. When is this the case? Quantum theory gives several pragmatic answers. Ref. 66 The first criterion for compositeness is somewhat strange: an object is compos- ite when its gyromagnetic ratio is different from the one predicted by quantum Page 189 electrodynamics. The gyromagnetic ratio 𝛾 – not to be confused with the relativistic dilation factor – is defined as the ratio between the magnetic moment 𝑀 and the Motion Mountain – The Adventure of Physics angular momentum 𝐿: 𝑀 = 𝛾𝐿 . (48) Challenge 77 e The gyromagnetic ratio 𝛾 is measured in units of s−1 T−1 , i.e., C/kg, and determines the energy levels of magnetic spinning particles in magnetic fields; it will reappear later in Vol. V, page 162 the context of magnetic resonance imaging. All candidates for elementary particles have spin 1/2. The gyromagnetic ratio for spin-1/2 particles of magnetic moment 𝑀 and mass 𝑚 can be written as 𝑀 𝑒 𝛾= =𝑔 . (49) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ℏ/2 2𝑚 The criterion for being elementary can thus be reduced to a condition on the value of the dimensionless number 𝑔, the so-called 𝑔-factor. (The expression 𝑒ℏ/2𝑚 is often called the magneton of the particle.) If the 𝑔-factor differs from the value predicted by quantum Page 189 electrodynamics for point particles – about 2.0 – the object is composite. For example, a 4 He+ helium ion has spin 1/2 and a 𝑔 value of 14.7 ⋅ 103 . Indeed, the radius of the helium ion is 3 ⋅ 10−11 m, obviously a finite value, and the ion is a composite entity. For the proton, one measures a 𝑔-factor of about 5.6. Indeed, experiments yield a finite proton radius of about 0.9 fm and show that it contains several constituents. The neutron, which has a magnetic moment despite being electrically neutral, must therefore be composite. Indeed, its radius is approximately the same as that of the proton. Similarly, molecules, mountains, stars and people must be composite. According to this first criterion, the only elementary particles are leptons (i.e., electrons, muons, tauons and neutrinos), quarks, and intermediate bosons (i.e., photons, W-bosons, Z-bosons and Vol. V, page 162 gluons). More details on these particles will be revealed in the chapters on the nucleus. Another simple criterion for compositeness has just been mentioned: any object with a measurable size is composite. This criterion yields the same list of elementary particles as the first. Indeed, the two criteria are related. The simplest model for composite structures 108 4 the quantum description of matter Ref. 67 predicts that the 𝑔-factor obeys 𝑅 𝑔−2= (50) 𝜆C where 𝑅 is the radius and 𝜆 C = ℎ/𝑚𝑐 is the Compton wavelength of the system. This Challenge 78 e expression is surprisingly precise for helium-4 ions, helium-3, tritium ions and protons, Vol. V, page 342 as you may wish to check. The tables in Appendix B in the next volume make the same point. In short, the second criterion for compositeness is equivalent to the first. A third criterion for compositeness is more general: any object larger than its Compton length is composite. The argument is simple. An object is composite if one can detect internal motion, i.e., motion of some components. Now the action of any part with mass 𝑚part moving inside a composed system of size 𝑟 obeys 𝑆part < 2π 𝑟 𝑚part 𝑐 < π 𝑟 𝑚 𝑐 (51) Motion Mountain – The Adventure of Physics where 𝑚 is the mass of the composite object. On the other hand, following the principle of quantum theory, this action, to be observable, must be larger than ℏ/2. Inserting this condition, we find that for any composite object* ℏ 𝑟> . (52) 2π 𝑚 𝑐 The right-hand side differs only by a factor 4π2 from the so-called Compton (wave)length ℎ copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 𝜆= (53) 𝑚𝑐 of an object. Thus any object larger than its own Compton wavelength is composite; and any object smaller than the right-hand side of expression (52) is elementary. Again, only leptons, quarks and intermediate bosons passed the test. (For the Higgs boson discovered in 2012, the test has yet to be performed, but it is expected to comply as well.) All other objects are composite. In short, this third criterion produces the same list as the previous Challenge 80 e ones. Can you explain why? A fourth criterion for compositeness is regularly cited by Steven Weinberg: a particle is elementary if it appears in the Lagrangian of the standard model of particle physics, Vol. V, page 261 i.e., in the description of the fundamental building blocks of nature. Can you show that Challenge 81 s this criterion follows from the previous ones? Interestingly, we are not yet finished with this topic. Even stranger statements about Vol. VI, page 313 compositeness will appear when gravity is taken into account. Just be patient: it is worth it. Challenge 79 ny * Can you find the missing factor of 2? And is the assumption that the components must always be lighter than the composite a valid one? 4 the quantum description of matter 109 Curiosities and fun challenges ab ou t quantum motion of mat ter “ Die meisten Physiker sind sehr naiv, sie glauben immer noch an wirkliche Wellen oder ” Teilchen.* Anton Zeilinger Take the sharpest knife edge or needle tip you can think of: the quantum of action implies that their boundaries are not sharp, but fuzzy, like the boundaries of clouds. Take the hardest or most solid object you can think of, such as diamond or a block of tungsten: the quantum of action implies that its surface is somewhat soft. All experiments confirm these statements. Nothing in nature is really sharp or really solid. Quantum physics thus disagrees with several ideas of the ancient Greek atomists. ∗∗ Do hydrogen atoms exist? Most types of atom have been imaged with microscopes, pho- Motion Mountain – The Adventure of Physics tographed under illumination, levitated one by one, and even moved with needles, one by one, as the picture on page 344 in volume I shows. Researchers have even moved single Ref. 68 atoms by using laser beams to push them. However, not a single one of these experi- ments has measured or imaged hydrogen atoms. Is that a reason to doubt the existence Challenge 82 s of hydrogen atoms? Taking this not-so-serious discussion seriously can be a lot of fun. ∗∗ Is the wave function ’real’? More precisely, is the wave function really a cloud? Some physicists still doubt this. This dying group of physicists, often born around the middle of the twentieth century, have heard so often – incorrectly and usually from questionable copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net authorities – that a wave function has no reality that they stopped asking and answering Challenge 83 e the simplest questions. To dispel their doubts, ask them whether they have a non-zero height or whether they think that atoms are round. If they agree, they have admitted that wave functions have some sort of reality. All everyday objects are made of elementary particles that are so unmeasurably small that we can call them point-like. Therefore, the size, surface area and volume of all everyday objects are exclusively due to wave func- tions. Every length, area and volume is a proof that wave functions have some sort of reality. ∗∗ Two observables can commute for two different reasons: either they are very similar – such as the coordinates 𝑥 and 𝑥2 – or they are very different – such as the coordinate 𝑥 Challenge 84 d and the momentum 𝑝𝑦 . Can you give an explanation for this? ∗∗ Space and time translations commute. Why then do the momentum operator and the Challenge 85 ny Hamiltonian not commute in general? ∗∗ * ‘Most physicists are very naive; they still believe in real waves or real particles.’ Anton Zeilinger, physicist at the University of Vienna, is well-known for his experiments on quantum mechanics. 110 4 the quantum description of matter F I G U R E 55 A special potential well that does not disturb a wave function. Colour indicates phase. (QuickTime ﬁlm © Bernd Thaller) Motion Mountain – The Adventure of Physics There exist special potentials that have no influence on a wave function. Figure 55 shows an example. This potential has reflection coefficient zero for all energies; the scattered wave has no reflected part. The mathematical reason is fascinating. The potential well has the shape of a soliton of the Korteweg–de Vries equation; this equation is related to the Schrödinger equation. ∗∗ Any bound system in a non-relativistic state with no angular momentum obeys the Ref. 69 relation copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 9ℏ2 ⟨𝑟2 ⟩ ⟨𝑇⟩ ⩾ , (54) 8𝑚 where 𝑚 is the reduced mass and 𝑇 the kinetic energy of the components, and 𝑟 is the size of the system. Can you deduce this result, and check it for the ground state of hydrogen? Challenge 86 s ∗∗ In high school, it often makes sense to visualize electron wave functions as a special type of fluid-like matter, called electronium, that has a negative charge density. In this visual- ization, an atom is a positive nucleus surrounded by an electronium cloud. Deforming the electronium cloud around a nucleus requires energy; this happens when a photon of the correct frequency is absorbed, for example. When atoms of the right kind approach each other, the electronium clouds often form stable bridges – chemical bonds. ∗∗ Quantum theory allows for many unusual bound states. Usually we think of bound states as states of low energy. But there are situations in which bound states arise due to external Vol. I, page 319 forcing with oscillating potentials. We encountered such a situation in classical physics: the vertically driven, upside-down pendulum that remain vertical despite being unstable. Similar situations also occur in quantum physics. Examples are Paul traps, the helium Ref. 70 atom, negative ions, Trojan electrons and particle accelerators. 4 the quantum description of matter 111 ∗∗ One often reads that the universe might have been born from a quantum fluctuation. Challenge 87 s Can you explain why this statement make no sense? A summary on motion of mat ter quantons In summary, the motion of massive quantons, i.e., of quantum matter particles, can be described in two ways: — At high magnification, quantum matter particles are described by wave functions that move like advancing, rotating and precessing clouds of arrows. The local cloud orienta- tion, or local phase, follows a wobbling motion. The square of the wave function, i.e., the density of the cloud, is the probability for finding the particle at a given spot. — Seen from far away, at low magnification, a moving massive quantum particle behaves as a single advancing, rotating and precessing arrow. The details of the rotation and precession of the arrow depend on the energy and momentum of the particle and the Motion Mountain – The Adventure of Physics potential it is subjected to. The arrow is a probability amplitude: the squared length of the arrow is the probability to observe the particle. If a particle can get from a starting point to a final point in several ways, the probability amplitudes for each way add up. The single rotating arrow results from a cloud average. The single arrow combines particle and wave properties. A full rotation of the arrow corresponds to the quantum of action ℏ. This central feature implies that a non-relativistic particle whose spin can be neglected follows the Schrödinger equation, and that a relativistic electron follows the Dirac equation. The Dirac equation agrees with all known experiments. In particular, copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net the Dirac equation describes all of materials science, chemistry and biology, as we will find out. To continue with the greatest efficiency on our path across quantum physics, we ex- plore three important topics: the indistinguishability of particles of the same kind, the spin of quantum particles, and the meaning of probabilities. Chapter 5 PE R M U TAT ION OF PA RT IC L E S – A R E PA RT IC L E S L I K E G LOV E S ? W hy are we able to distinguish twins from each other? Why can we distinguish hat looks alike, such as a copy from an original? Most of us are convinced that henever we compare an original with a copy, we can find a difference. This con- viction turns out to be correct also in the quantum domain, but the conclusion is not Motion Mountain – The Adventure of Physics straightforward. Think about any method that allows you to distinguish objects: you will find that it Challenge 88 s runs into trouble for point-like particles. Therefore, in the quantum domain something must change about our ability to distinguish particles and objects. We could argue that differences between an original object and a copy can always be made to disappear: it should be sufficient to use the same number and type of atoms. In fact, the quantum of action shows that this is not sufficient, even though all atoms of the same type are indeed indistinguishable copies of each other! In the following we explore the most important consequences on motion of the indistinguishability of atoms and of the distinguishability of macroscopic objects. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Distinguishing macroscopic objects A number of important properties of objects are highlighted by studying a combinatorial puzzle: the glove problem. It asks: How many surgical gloves (for the right hand) are necessary if 𝑚 doctors need to operate 𝑤 patients in a hygienic way, so that nobody gets in contact with the body fluids of anybody else? The same problem also appears in other settings. For example, it also applies to com- Ref. 71 puters, interfaces and computer viruses or to condoms, men and women – and is then called the condom problem. To be clear, the optimal number of gloves is not the product 𝑚𝑤. In fact, the problem has three subcases. Challenge 89 s — The simple case 𝑚 = 𝑤 = 2 already provides the most important ideas needed. Are you able to find the optimal solution and procedure? Challenge 90 e — In the case 𝑤 = 1 and 𝑚 odd, the solution is (𝑚 + 1)/2 gloves. The corresponding expression (𝑤 + 1)/2 holds for the case 𝑚 = 1 and 𝑤 odd. This is the optimal solution, as you can easily check yourself. Ref. 72 — A solution with a simple procedure for all other cases is given by ⌈2𝑤/3+𝑚/2⌉ gloves, where ⌈𝑥⌉ means the smallest integer greater than or equal to 𝑥. For example, for two 5 permutation of particles 113 doctors and three patients this gives only three gloves. (However, this formula does Challenge 91 e not always give the optimal solution; better values exist in certain subcases.) Enjoy working on the puzzle. You will find that three basic properties of gloves determine the solution. First, gloves have two sides, an interior and an exterior one, that can be distinguished from each other. Secondly, gloves turned inside out exchange left and right and can thus be distingusihed from gloves that are not reversed. Thirdly, gloves can be distinguished from each other. Now we come back to our original aim: Do the three basic properties of gloves also apply to quantum particles? We will explore the issue of double-sidedness of quantum Vol. VI, page 114 particles in the last part of our mountain ascent. The question whether particles can be turned inside out will be of importance for their description and their motion. We will also explore the difference between right- and left-handed particles, though in the next Vol. V, page 245 part of our adventure. In the present chapter we concentrate on the third issue, namely whether objects and particles can always be distinguished from copies. We will find that elementary particles do not behave like gloves – but in a much more surprising manner. Motion Mountain – The Adventure of Physics In everyday life, distinction of macroscopic objects can be achieved in two ways. On the one hand, we are able to distinguish objects – or people – from each other because they differ in their intrinsic properties, such as their mass, colour, size or shape. On the other hand, we are able to distinguish objects even if they have the same intrinsic prop- erties. Any game of billiard shows us that by following the path of each ball, we can distinguish it from the other balls. In short, we can distinguish objects with identical properties also using their state. The state of a billiard ball is given by its position, its linear and its angular momentum. We are able to distinguish two identical billiard balls because the measurement error for copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net the position of each ball is much smaller than the size of the ball itself. The different states of two billiard balls allow us to track each ball. However, in the microscopic domain, this is not possible! Let us take two atoms of the same type. Two such atoms have exactly the same intrinsic properties. To distinguish them in collisions, we would need to keep track of their motion. But due to the quantum of action and the ensuing indeterminacy rela- tion, we have no chance to achieve this. In fact, a simple experiment from the nineteenth century showed that even nature itself is not able to do it! This profound result was dis- covered studying systems which incorporate a large number of colliding atoms of the same type: gases. Distinguishing atoms Vol. I, page 402 What is the entropy of a gas? The calculation of the entropy 𝑆 of a simple gas, made of 𝑁 simple particles* of mass 𝑚 moving in a volume 𝑉, gives 𝑆 𝑉 3 ln 𝛼 = ln [ 3 ] + + . (55) 𝑘𝑁 Λ 2 𝑁 Here, 𝑘 is the Boltzmann constant, ln the natural logarithm, 𝑇 the temperature, and Λ = √2πℏ2 /𝑚𝑘𝑇 is the thermal wavelength (approximately the de Broglie wavelength of the * Particles are simple if they are fully described by their momentum and position; atoms are simple particles. Molecules are not simple, as they are describe also by their orientation. 114 5 permutation of particles F I G U R E 56 Willard Gibbs (1839 –1903) particles making up the gas). In this result, the pure number 𝛼 is equal to 1 if the particles are distinguishable like billiard balls, and equal to 1/𝑁! if they are not distinguishable at Challenge 92 e all. Measuring the entropy of a simple gas thus allows us to determine 𝛼 and therefore to Motion Mountain – The Adventure of Physics test experimentally whether particles are distinguishable. It turns out that only the second case, 𝛼 = 1/𝑁!, describes nature. We can easily check Challenge 93 e this without even performing the measurement: only in the second case does the entropy of two volumes of identical gas add up.* The result, often called Gibbs’ paradox,** thus Ref. 73 proves that the microscopic components of matter are indistinguishable: in a system of quantum particles – be they electrons, protons, atoms or small molecules – there is no way to say which particle is which. Indistinguishability of particles is thus an experimental property of nature. It holds without exception. For example, when radioactivity was discovered, people thought that copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net it contradicted the indistinguishability of atoms, because decay seems to single out cer- tain atoms compared to others. But quantum theory then showed that this is not the case and that even atoms and molecules are indistinguishable. Since ℏ appears in the expression for the entropy, indistinguishability is a quantum effect. Indeed, indistinguishability plays no role if quantum effects are negligible, as is the case for billiard balls. Nevertheless, indistinguishability is important in everyday life. We will find out that the properties of everyday matter – plasma, gases, liquids and solids – would be completely different without indistinguishability. For example, we will discover that without it, knifes and swords would not cut. In addition, the soil would not carry us; we would fall right through it. To illuminate the issue in more detail, we explore the following question. * Indeed, the entropy values observed by experiment, for a monoatomic gas, are given by the so-called Challenge 94 d Sackur–Tetrode formula 𝑆 𝑉 5 = ln [ ]+ (56) 𝑘𝑁 𝑁Λ3 2 which follows when 𝛼 = 1/𝑁! is inserted above. It was deduced independently by the German physicist Otto Sackur (1880–1914) and the Dutch physicist Hugo Tetrode (1895–1931). Note that the essential parameter is the ratio between 𝑉/𝑁, the classical volume per particle, and Λ3 , the de Broglie volume of a quantum particle. ** Josiah Willard Gibbs (1839–1903), US-American physicist who was, with Maxwell and Planck, one of the three founders of statistical mechanics and thermodynamics; he introduced the concept of ensemble and the term thermodynamic phase. 5 permutation of particles 115 m m F I G U R E 57 Identical objects with crossing paths. Why d oes indistinguishability appear in nature? Motion Mountain – The Adventure of Physics Take two quantum particles with the same mass, the same composition and the same shape, such as two atoms of the same kind. Imagine that their paths cross, and that they approach each other to small distances at the crossing, as shown in Figure 57. In a gas, both a collision of atoms or a near miss are examples. Now, all experiments ever per- formed show that at small distances it is impossible to say whether the two quantons have switched roles or not. ⊳ It is impossible in a gas to follow quantum particles moving around and to determine which one is which. Tracking colliding quantons is impossible. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net The impossibility to distinguish nearby particles is a direct consequence of the quantum of action ℏ. For a path that brings two approaching particles very close to each other, a role switch requires only a small amount of change, i.e., only a small (physical) action. However, we know that there is a smallest observable action in nature. Keeping track of each quantum particle at small distances would require action values smaller than the quantum of action. The existence of the quantum of action thus makes it impossible to keep track of quantum particles when they come too near to each other. Any description of systems with several quantons must thus take into account that after a close encounter, it is impossible to say which quanton is which. If we remember that quantum theory describes quantons as clouds, the indistin- guishability appears even more natural. Whenever two clouds meet and depart again, it is impossible to say which cloud is which. On the other hand, if two particles are kept distant enough, one does have an effective distinguishability; indistinguishability thus appears only when the particles come close. In short, indistinguishability is a natural, unavoidable consequence of the existence of a smallest action value in nature. This result leads us straight away to the next question: C an quantum particles be counted? In everyday life, we can count objects because we can distinguish them. Since quantum particles cannot always be distinguished, we need some care in determining how to count 116 5 permutation of particles them. The first step in counting particles is the definition of what is meant by a situation without any particle at all. This seems an easy thing to do, but later on we will encounter situations where already this step runs into difficulties. In any case, the first step of count- ing is thus the specification of the vacuum. Any counting method requires that the situ- ation without particles is clearly separated from situations with particles. The second step necessary for counting is the specification of an observable useful for determining quantum particle number. The easiest way is to choose one of those conserved quantum numbers that add up under composition, such as electric charge. Counting itself is then performed by measuring the total charge and dividing by the unit charge. In everyday life, the weight or mass is commonly used as observable. However, it cannot be used generally in the quantum domain, except for simple cases. For a large number of particles, the interaction energy will introduce errors. For very large particle numbers, the gravitational binding energy will do so as well. But above all, for transient phenomena, unstable particles or short measurement times, mass measurements reach Motion Mountain – The Adventure of Physics their limits. In short, even though counting stable atoms through mass measurements works in everyday life, the method is not applicable in general; especially at high particle energies, it cannot be applied. Counting with the help of conserved quantum numbers has several advantages. First of all, it works also for transient phenomena, unstable particles or short measurement times. Secondly, it is not important whether the particles are distinguishable or not; Vol. II, page 72 counting always works. Thirdly, virtual particles are not counted. This is a welcome Vol. V, page 127 state of affairs, as we will see, because for virtual particles, i.e., particles for which Vol. II, page 72 𝐸2 ≠ 𝑝2 𝑐2 + 𝑚2 𝑐4 , there is no way to define a particle number anyway. Using a conserved copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net quantity is indeed the best particle counting method possible. The side effect of counting with the help of quantum numbers is that antiparticles count negatively! Also this consequence is a result of the quantum of action. We saw above that the quantum of action implies that even in vacuum, particle–antiparticle pairs are observed at sufficiently high energies. As a result, an antiparticle must count as minus one particle. In other words, any way of counting quantum particles can produce an error due to this effect. In everyday life this limitation plays no role, as there is no antimatter around us. The issue does play a role at higher energies, however. It turns out that there is no general way to count the exact number of particles and antiparticles separately; only the sum can be defined. In short, quantum theory shows that particle counting is never perfect. In summary, nature does provide a way to count quantum particles even if they cannot be distinguished, though only for everyday, low energy conditions; due to the quantum of action, antiparticles count negatively. Antiparticles thus provide a limit to the counting of particles at high energies, when the mass–energy equivalence becomes important. What is permu tation symmetry? Since quantum particles are countable but indistinguishable, there exists a symmetry of nature for systems composed of several identical quantons. Permutation symmetry, also called exchange symmetry, is the property of nature that observations are unchanged un- der exchange of identical particles. Permutation symmetry forms one of the four pil- 5 permutation of particles 117 lars of quantum theory, together with space-time symmetry, gauge symmetry and the not yet encountered renormalization symmetry. Permutation symmetry is a property of composed systems, i.e., of systems made of many (identical) subsystems. Only for such systems does indistinguishability play a role. In other words, ‘indistinguishable’ is not the same as ‘identical’. Two quantum particles of the same type are not the same; they are more like exact copies of each other. On the other hand, everyday life experience shows us that two copies can always be dis- tinguished under close inspection, so that the term ‘copy’ is not fully appropriate either. ⊳ Quantons, quantum particles, are countable and completely indistinguish- able.* Quantum particles are perfect copies of each other. Being perfect copies, not even nature can distinguish particles; as a result, permutation symmetry appears. In the next chapter, we will discover that permutation is partial rotation. Permutation Motion Mountain – The Adventure of Physics Challenge 95 e symmetry thus is a symmetry under partial rotations. Can you find out why? Indistinguishabilit y and wave function symmetry The indistinguishability of quantum particles leads to important conclusions about the description of their state of motion. This happens because it is impossible to formulate a description of motion that includes indistinguishability right from the start. (Are you Challenge 96 s able to confirm this?) We need to describe a 𝑛-particle state with a state Ψ1...𝑖...𝑗...𝑛 which assumes that distinction is possible, as expressed by the ordered indices in the notation, and we introduce the indistinguishability afterwards. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Indistinguishability, or permutation symmetry, means that the exchange of any two quantum particles results in the same physical observations.** Now, two quantum states have the same physical properties if they differ at most by a phase factor; indistinguishab- ility thus requires Ψ1...𝑖...𝑗...𝑛 = e𝑖𝛼 Ψ1...𝑗...𝑖...𝑛 (57) for some unknown angle 𝛼. Applying this expression twice, by exchanging the same couple of indices again, allows us to conclude that e2𝑖𝛼 = 1. This implies that Ψ1...𝑖...𝑗...𝑛 = ± Ψ1...𝑗...𝑖...𝑛 , (58) in other words, a wave function is either symmetric or antisymmetric under exchange of indices. (We can also say that the eigenvalue for the exchange operator is either +1 or −1.) ⊳ Quantum theory thus predicts that quantum particles can be indistinguish- * The word ‘indistinguishable’ is so long that many physicists sloppily speak of ‘identical’ particles never- theless. Take care. ** We therefore have the same situation that we encountered already several times: an overspecification of the mathematical description, here the explicit ordering of the indices, implies a symmetry of this description, which in our case is a symmetry under exchange of indices, i.e., under exchange of particles. 118 5 permutation of particles able in one of two distinct ways.* ⊳ Particles corresponding to symmetric wave functions – those which trans- form under particle exchange with a ‘+’ in equation (58) – are called** bo- sons. ⊳ Particles corresponding to antisymmetric wave functions – those which transform under particle exchange with a ‘−’ in equation (58) – are called*** fermions. Experiments show that the exchange behaviour depends on the type of particle. Photons are found to be bosons. On the other hand, electrons, protons and neutrons are found to be fermions. Also about half of the atoms are found to behave as bosons (at moderate energies), the other half are fermions. To determine they type of atom, we need to take into account the spin of the electron and that of the nucleus. In fact, a composite of an even number of fermions (at moderate energies) – or of Motion Mountain – The Adventure of Physics any number of bosons (at any energy) – turns out to be a boson; a composite of an odd number of fermions is (always) a fermion. For example, 4 He is a boson, 3 He a fermion. Also the natural isotopes 23 Na, 41 K, 85 Rb, 87 Rb and 133 Cs are bosons, because they have odd numbers of electrons and of nucleons; in contrast, 40 K and 134 Cs are fermions (and, in this case, also radioactive). The behaviour of photons A simple experiment, shown in Figure 58, allows observing an important aspect of photon behaviour. Take a source that emits two indistinguishable photons, i.e., two copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net photons of identical frequency and polarization, at the same time. The photon pair is therefore in an entangled state. In the laboratory, such a source can be realized with a down-converter, a material that converts a photon of frequency 2𝑓 into two photons of frequency 𝑓. The two entangled photons, after having travelled exactly the same distance, are made to enter the two sides of an ideal beam splitter (for example, a half-silvered mir- ror). Two detectors are located at the two exits of the beam splitter. Experiments show Ref. 75 that both photons are always detected together on the same side, and never separately on * This conclusion applies to three-dimensional space. In two dimensions there are more possibilities. Such possibilities have been and partly still are topic of research. ** ‘Bosons’ are named after the physicist Satyenra Nath Bose (b. 1894 Calcutta, d. 1974 Calcutta) who first Ref. 74 described the statistical properties of photons. The work was later expanded by Albert Einstein, so that one speaks of Bose–Einstein statistics. *** The term ‘fermion’ is derived from the name of the physicist and Nobel Prize winner Enrico Fermi (b. 1901 Rome, d. 1954 Chicago) famous for his all-encompassing genius in theoretical and experimental physics. He mainly worked on nuclear and elementary particle physics, on spin and on statistics. For his experimental work he was called ‘quantum engineer’. He is also famous for his lectures, which are still published in his own hand-writing, and his brilliant approach to physical problems. Nevertheless, his highly deserved Nobel Prize was one of the few cases in which the prize was given for a discovery which turned out to be incorrect. He left Italy because of the bad treatment his Jewish wife was suffering and emigrated to the USA. Fermi worked on the Manhattan project that built the first atomic bombs. After the Second World War, he organized one of the best physics department in the world, at the University of Chicago, where he was admired by everybody who worked with him. 5 permutation of particles 119 detectors mirrors beam source splitter F I G U R E 58 two photons of possible light Two-photon emission same frequency f paths and interference: two one photon of and polarization indistinguishable frequency 2f photons are always found arriving together, at the same detector. Motion Mountain – The Adventure of Physics 4He shows bunching classical prediction classical prediction copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 3He shows anti-bunching F I G U R E 59 Bunching and antibunching of 3 He and 4 He helium!bunching atoms: the measurement result, the detector and the experiment (from atomoptic.iota.u-psud.fr/research/helium/helium.html, photo © Denis Boiron, Jerome Chatin). opposite sides. This happens because the two options where one of the photons is trans- mitted and the other reflected interfere destructively. (The discussion mentioned above Page 59 applies also here: despite two photons being involved, also in this case, when investigat- ing the details, only one photon interferes with itself.) The experiment shows that photons are bosons. Indeed, in the same experiment, fer- mions behave in exactly the opposite way; two fermions are always detected separately Ref. 76 on opposite sides, never together on the same side. 120 5 permutation of particles F I G U R E 60 Picturing particles as localized excitations (left) or clouds (right). Bunching and antibunching Another way to test the exchange character of a particle is the Hanbury Brown–Twiss ex- Page 53 periment described earlier on. First of all, this beautiful experiment shows that quantum particles behave differently than classical particles. In addition, compared to classical particles, fermions show antibunching – because of Pauli’s exclusion principle – and Motion Mountain – The Adventure of Physics bosons show bunching. Hanbury Brown and Twiss performed the experiment with photons, which are bosons. Ref. 77 In 2005, a French–Dutch research collaboration performed the experiment with atoms. By using an extremely cold helium gas at 500 nK and a clever detector principle, they were able to measure the correlation curves typical for the effect. The results, shown in Figure 59, confirm that 3 He is a fermion and 4 He is a boson, as predicted from the composition rule of quantum particles. The energy dependence of permu tation symmetry copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net If experiments force us to conclude that nobody, not even nature, can distinguish between two particles of the same type, we deduce that they do not form two separ- ate entities, but some sort of unity. Our naive, classical sense of particle as a separate entity from the rest of the world is thus an incorrect description of the phenomenon of ‘particle’. Indeed, no experiment can track particles with identical intrinsic properties in such a way that they can be distinguished with certainty. This impossibility has been checked experimentally with all elementary particles, with nuclei, with atoms and with numerous molecules. How does this fit with everyday life, i.e., with classical physics? Photons do not worry us much here. Let us focus the discussion on matter particles. We know to be able to distinguish electrons by pointing to different wires in which they flow; also, we can dis- tinguish our fridge, with its electrons and atoms, from that of our neighbour. While the quantum of action makes distinction impossible, everyday life does allow it. The simplest explanation for both observations is to imagine a microscopic particle, especially an elementary one, as a bulge, i.e., as a localized excitation of the vacuum, or as a tiny cloud. Figure 60 shows two such bulges and two clouds representing particles. It is evident that if particles are too near to each other, it makes no sense to distinguish them; we cannot say any more which is which. The bulge image shows that either for large distances or for high potential walls sep- arating them, distinction of identical particles does become possible. In such situations, measurements allowing us to track particles independently do exist – as we know from 5 permutation of particles 121 everyday life. In other words, we can specify a limit energy at which permutation sym- metry of objects or particles separated by a distance 𝑑 becomes important. It is given by 𝑐ℏ 𝐸= . (59) 𝑑 Challenge 97 e Are you able to confirm the expression? For example, at everyday temperatures we can distinguish atoms inside a solid from each other, since the energy so calculated is much higher than the thermal energy of atoms. To have fun, you might want to determine at Challenge 98 e what energy two truly identical human twins become indistinguishable. Estimating at what energies the statistical character of trees or fridges will become apparent is then straightforward. To sum up, in daily life we are able to distinguish objects and thus people for two reasons: because they are made of many parts, and because we live in a low energy envir- onment. The bulge image of particles purveys the idea that distinguishability exists for Motion Mountain – The Adventure of Physics objects in everyday life but not for particles in the microscopic domain. The energy issue immediately adds a new aspect to the discussion. How can we de- scribe fermions and bosons in the presence of virtual particles and of antiparticles? Indistinguishabilit y in quantum field theory Quantum field theory, as we will see in the next volume, simply puts the cloudy bulge idea of Figure 60 into mathematical language. A situation without any bulge is called vacuum state. Quantum field theory describes all particles of a given type as excitations of a single fundamental field. Particles are indistinguishable because each particle is an copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net excitation of the same basic substrate and each excitation has the same properties. A situation with one particle is then described by a vacuum state acted upon by a creation operator. Adding a second particle is described by adding a second creation operator, and subtracting a particle by adding a annihilation operator; the latter turns out to be the adjoint of the former. Quantum field theory studies how creation and annihilation operators must behave to describe observations.* It arrives at the following conclusions: — Field operators for particles with half-integer spin are fermions and imply (local) an- ticommutation. — Field operators for particles with with integer spin are bosons and imply (local) com- mutation. — For all field operators, the commutator, respectively anticommutator, taken at two points with space-like separations, vanishes. * Whenever the relation [𝑏, 𝑏† ] = 𝑏𝑏† − 𝑏† 𝑏 = 1 (60) † holds between the creation operator 𝑏 and the annihilation operator 𝑏, the operators describe a boson. The dagger can thus be seen as describing the operation of adjoining; a double dagger is equivalent to no dagger. If the operators for particle creation and annihilation anticommute {𝑑, 𝑑† } = 𝑑𝑑† + 𝑑† 𝑑 = 1 (61) they describe a fermion. The so defined bracket is called the anticommutator bracket. 122 5 permutation of particles — Antiparticles of fermions are fermions, and antiparticles of bosons are bosons. — Virtual particles behave under exchange like their real counterparts. These connections are at the basis of quantum field theory. They describe how quantons behave under permutation. But why are quantum particles identical? Why are all electrons identical? Lead by experiment, quantum field theory describes electrons as identical excitations of the va- cuum, and as such as identical by construction. Of course, this answer is not really sat- isfying. We will find a better one only in the final part of our mountain ascent. How accurately is permu tation symmetry verified? Are electrons perfect fermions? In 1990, a simple but effective experiment testing their Ref. 78 fermion behaviour was carried out by Ramberg and Snow. They sent an electric current of 30 A through a copper wire for one month and looked for X-ray emission. They did not find any. They concluded that electrons are always in an antisymmetric state, with a Motion Mountain – The Adventure of Physics symmetric component of less than 2 ⋅ 10−26 (62) of the total state. In short, electrons are always in an antisymmetric state: they are fermi- ons. The reasoning behind this elegant experiment is the following. If electrons would not always be fermions, every now and then an electron could fall into the lowest energy level of a copper atom, leading to X-ray emission. The lack of such X-rays implies that electrons are fermions to a very high accuracy. X-rays could be emitted only if they were copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net bosons, at least part of the time. Indeed, two electrons, being fermions, cannot be in the same quantum state: this restriction is called the Pauli exclusion principle. It applies to all fermions and is the topic of the next chapter. C opies, clones and gloves Can classical systems be indistinguishable? They can: large molecules are examples – provided they are made of exactly the same isotopes. Can large classical systems, made of a mole or more particles be indistinguishable? This simple question effectively asks whether a perfect copy, or (physical) clone, of a physical system is possible. It could be argued that any factory for mass-produced goods, such as one producing shirt buttons or paper clips, shows that copies are possible. But the appearance is deceiv- ing. Seen under a microscope, there is usually some difference. Is this always the case? In 1982, the Dutch physicist Dennis Dieks and independently, the US-American physicists Ref. 79 Wootters and Zurek, published simple proofs that quantum systems cannot be copied. This is the famous no-cloning theorem. A copying machine is a machine that takes an original, reads out its properties and produces a copy, leaving the original unchanged. This definition seems straightforward. However, we know that if we extract information from an original, we have to interact with it. As a result, the system will change at least by the quantum of action. We thus expect that due to quantum theory, copies and originals can never be identical.* * This seems to provide a solution against banknote forgeries. In fact, Stephen Wiesner proposed to use 5 permutation of particles 123 Quantum theory indeed shows that copying machines are impossible. A copying ma- chine is described by an operator that maps the state of an original system to the state of the copy. In other words, a copying machine is linear. This linearity leads to a problem. Simply stated, if a copying machine were able to copy originals either in state |𝐴⟩ or in state |𝐵⟩, it could not work if the state of the original were a superposition |𝐴⟩ + |𝐵⟩. Let us see why. A copy machine is a device described by an operator 𝑈 that changes the starting state |𝑠⟩c of the copy in the following way: — If the original is in state |𝐴⟩, a copier acts on the copy |𝑠⟩c as 𝑈|𝐴⟩|𝑠⟩c = |𝐴⟩|𝐴⟩c . (63) — If the original is in state |𝐵⟩, a copier acts on the copy |𝑠⟩c as 𝑈|𝐵⟩|𝑠⟩c = |𝐵⟩|𝐵⟩c . (64) Motion Mountain – The Adventure of Physics As a result of these two requirements, an original in the state |𝐴 + 𝐵⟩ is treated by the copier as 𝑈|𝐴 + 𝐵⟩|𝑠⟩c = |𝐴⟩|𝐴⟩c + |𝐵⟩|𝐵⟩c . (65) This is in contrast to what we want, which would be 𝑈wanted|𝐴 + 𝐵⟩|𝑠⟩c = (|𝐴⟩ + |𝐵⟩)(|𝐴⟩c + |𝐵⟩c ) . (66) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net In other words, a copy machine cannot copy a state completely.* This is the so-called no-cloning theorem. The impossibility of copying is implicit in quantum theory. If we were able to clone systems, we could measure a variable of a system and a second variable on its copy. We would be thus able to beat the indeterminacy relation in both copies. This is impossible. In short, copies are always imperfect. The lack of quantum mechanical copying machines is disappointing. Such science fiction machines could be fed with two different inputs, such as a lion and a goat, and produce a superposition: a chimaera. Quantum theory shows that all these imaginary beings or situations cannot be realized. Other researchers then explored how near to perfection a copy can be, especially in the Ref. 81 case of classical systems. To make a long story short, these investigations show that also the copying or cloning of macroscopic systems is impossible. In simple words, copying machines do not exist. Copies can always be distinguished from originals if observations Ref. 80 quantum theory already in 1970; he imagined to use polarizations of stored single photons as bits of serial Challenge 99 s numbers. Can you explain why this cannot work? * The no-cloning theorem puts severe limitations on quantum computers, as computations often need cop- ies of intermediate results. The theorem also shows that faster-than-light communication is impossible in EPR experiments. In compensation, quantum cryptography becomes possible – at least in the laboratory. Indeed, the no-cloning theorem shows that nobody can copy a quantum message without being noticed. The specific ways to use this result in cryptography are the 1984 Bennett–Brassard protocol and the 1991 Ekert protocol. 124 5 permutation of particles are made with sufficient care. In particular, this is the case for biological clones; biological clones are identical twins born following separate pregnancies. They differ in their finger prints, iris scans, physical and emotional memories, brain structures, and in many other Challenge 100 s aspects. (Can you specify a few more?) In short, biological clones, like identical twins, are not copies of each other. In summary, everyday life objects such as photocopies, billiard balls or twins are al- ways distinguishable. There are two reasons: first, quantum effects play no role in every- day life, so that there is no danger of unobservable exchange; secondly, perfect clones of classical systems do not exist anyway, so that there always are tiny differences between any two objects, even if they look identical at first sight. Gloves, being classical systems, can thus always be distinguished. Summary As a consequence of the quantum of action ℏ, quantum particles are indistinguishable. This happens in one of two ways: they are either bosons or fermions. Not even nature is Motion Mountain – The Adventure of Physics able to distinguish between identical quantum particles. Despite the indistinguishability of quantons, the state of a physical system cannot be copied to a second system with the same particle content. Therefore, perfect clones do not exist in nature. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Chapter 6 ROTAT ION S A N D STAT I ST IC S – V I SUA L I Z I NG SPI N S pin is the observation that matter beams can be polarized: rays can be rotated. pin thus describes how particles behave under rotations. Particles are thus not imply point-like: quantum particles can rotate around an axis. This proper rotation Page 104 is called spin; like macroscopic rotation, spin is described by an angular momentum. Motion Mountain – The Adventure of Physics In the following, we recall that the spin of quantons is quantized in units of ℏ/2. Then we show a deep result: the value of spin determines whether a quantum particle, and any general quantum system, is a boson or a fermion. And we will show that spin is the rotation of quantons. Q uantum particles and symmetry Ref. 82 The general background for the appearance of spin was clarified by Eugene Wigner in 1939.** He started by recapitulating that any quantum particle, if elementary, must be- have like an irreducible representation of the set of all viewpoint changes. This set of view- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net point changes forms the symmetry group of flat space-time, the so-called inhomogeneous Lorentz group. Why? Vol. I, page 266 We have seen in the chapter on symmetry, in the first volume of this adventure, that the symmetry of any composite system leads to certain requirements for the components of the system. If the components do not follow these requirements, they cannot build a symmetric composite. We know from everyday life and precision experiments that all physical systems are symmetric under translation in time and space, under rotation in space, under boosts, and – in many cases – under mirror reflection, matter–antimatter exchange and motion reversal. We know these symmetries from everyday life; for example, the usefulness of what we call ‘experience’ in everyday life is simply a consequence of time translation symmetry. The set of all these common symmetries, more precisely, of all these symmetry transformations, is called the inhomogeneous Lorentz group. These symmetries, i.e., these changes of viewpoints, lead to certain requirements for the components of physical systems, i.e., for the elementary quantum particles. In math- Vol. I, page 266 ematical language, the requirement is expressed by saying that elementary particles must be irreducible representations of the symmetry group. ** Eugene Wigner (b. 1902 Budapest, d. 1995 Princeton), theoretical physicist, received the Nobel Prize in Physics in 1963. He wrote over 500 papers, many about various aspects of symmetry in nature. He was also famous for being the most polite physicist in the world. 126 6 rotations and statistics – visualizing spin Every textbook on quantum theory carries out this reasoning in systematic detail. Starting with the Lorentz group, one obtains a list of all possible irreducible representa- tions. In other words, on eobtains a list of all possible ways that elementary particles can behave. * Cataloguing the possibilities, one finds first of all that every elementary particle is described by four-momentum – no news so far – by an internal angular momentum, the spin, and by a set of parities. — Four-momentum results from the translation symmetry of nature. The momentum value describes how a particle behaves under translation, i.e., under position and time shift of viewpoints. The magnitude of four-momentum is an invariant property, given by the mass, whereas its orientation in space-time is free. — Spin results from the rotation symmetry of nature. The spin value describes how an object behaves under rotations in three dimensions, i.e., under orientation change of viewpoints.** The magnitude of spin is an invariant property, and its orientation has various possibilities with respect to the direction of motion. In particular, the spin of massive quantum particles behaves differently from that of massless quantum Motion Mountain – The Adventure of Physics particles. For massive quantum particles, the inhomogeneous Lorentz group implies that the invariant magnitude of spin is √𝐽(𝐽 + 1) ℏ, often written, by oversimplification, as 𝐽. It is thus customary to say and write ‘spin J’ instead of the cumbersome ‘spin √𝐽(𝐽 + 1) ℏ’. Since the value of the quantum number 𝐽 specifies the magnitude of the angular momentum, it gives the representation under rotations of a given particle type. The exploration shows that the spin quantum number 𝐽 can be any multiple of 1/2, i.e., it can take the values 0, 1/2, 1, 3/2, 2, 5/2, etc. As summarized in Table 4, experiments show that electrons, protons and neutrons have spin 1/2, the W and Z copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net particles spin 1 and helium atoms spin 0. In addition, the representation of spin 𝐽 is 2𝐽 + 1 dimensional, meaning that the spatial orientation of the spin has 2𝐽 + 1 possible values. For electrons, with 𝐽 = 1/2, there are thus two possibilities; they are usually called ‘up’ and ‘down’. Spin thus only takes discrete values. This is in contrast with linear momentum, whose representations are infinite dimensional and whose possible values form a continuous range. Also massless quantum particles are characterized by the value of their spin. It can take the same values as in the massive case. For example, photons and gluons have spin 1. For massless particles, the representations are one-dimensional, so that mass- less particles are completely described by their helicity, defined as the projection of the spin onto the direction of motion. Massless particles can have positive or negat- ive helicity, often also called right-handed and left-handed polarization. There is no other freedom for the orientation of spin in the massless case. — To complete the list of particle properties, the remaining, discrete symmetries of the inhomogeneous Lorentz group must be included. Since motion inversion, spatial par- ity and charge inversion are parities, each elementary particle has to be described by three additional numbers, called T, P and C, each of which can only take the values * To be of physical relevance for quantum theory, representations have to be unitary. The full list of irre- ducible and unitary representations of viewpoint changes thus provides the range of possibilities for any particle that wants to be elementary. ** The group of physical rotations is also called SO(3), since mathematically it is described by the group of Special Orthogonal 3 by 3 matrices. 6 rotations and statistics – visualizing spin 127 TA B L E 4 Particle spin as representation of the rotation group. Spin System Massive examples Massless examples [ℏ] unchanged after elementary composite elementary rotation by 0 any angle Higgs mesons, nuclei, none 𝑎 boson atoms 1/2 2 turns 𝑒, 𝜇, 𝜏, 𝑞, nuclei, atoms, none, as neutrinos have a tiny mass 𝜈𝑒 , 𝜈𝜇 , 𝜈𝜏 molecules, radicals 1 1 turn W, Z mesons, nuclei, photon 𝛾, gluon 𝑔 atoms, molecules, toasters 3/2 2/3 turn none 𝑎 baryons, nuclei, none 𝑎 Motion Mountain – The Adventure of Physics atoms 𝑏 2 1/2 turn none nuclei ‘graviton’ 5/2 2/5 turn none nuclei none 3 1/3 turn none nuclei 𝑐 none 𝑐 𝑐 𝑐 𝑐 etc. etc. etc. etc. none possible 𝑎. Supersymmetry, a symmetry conjectured in the twentieth century, predicts elementary particles in these and other boxes. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 𝑏. The graviton has not yet been observed. 𝑐. Nuclei exist with spins values up to at least 101/2 and 51 (in units of ℏ). Ref. 83 +1 or −1. Being parities, these numbers must be multiplied to yield the value for a composed system. In short, the symmetries nature lead to the classification of all elementary quantum particles by their mass, their momentum, their spin and their P, C and T parities. Types of quantum particles The spin values observed for all quantum particles in nature are given in Table 4. The parities and all known intrinsic properties of the elementary particles are given in Table 5. Spin and parities together are called quantum numbers. All other intrinsic properties of quantons are related to interactions, such as mass, electric charge or isospin, and we will Vol. V, page 162 explore them in the next volume. 128 6 rotations and statistics – visualizing spin TA B L E 5 Elementary particle properties. Particle Mass 𝑚 𝑎 Lifetime 𝜏 Isospin 𝐼, Charge, Lepton or energy spin 𝐽, 𝑐 isospin, & width, 𝑏 parity 𝑃, strange- baryon 𝑒 main decay charge ness, 𝑐 num- modes parity 𝐶 charm, bers beauty, 𝑑 𝐿𝐵 topness: 𝑄𝐼𝑆𝐶𝐵𝑇 Elementary radiation (bosons) photon 𝛾 0 (<10−53 kg) stable 𝐼(𝐽𝑃𝐶 ) = 000000 0, 0 0, 1(1−−) 𝑊± 80.398(25) GeV/𝑐2 2.124(41) GeV 𝐽 = 1 ±100000 0, 0 67.60(27) % hadrons, Motion Mountain – The Adventure of Physics 32.12(36) % 𝑙+ 𝜈 𝑍 91.1876(21) GeV/𝑐2 2.65(2) ⋅ 10−25 s 𝐽 = 1 000000 0, 0 or 2.4952(23) GeV/𝑐2 69.91(6) % hadrons, 10.0974(69) % 𝑙+ 𝑙− gluon 0 stable 𝐼(𝐽𝑃 ) = 0(1− ) 000000 0, 0 Elementary matter (fermions): leptons electron 𝑒 9.109 382 15(45) ⋅ > 13 ⋅ 1030 s 𝐽 = 12 −100 000 1, 0 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 10−31 kg = 81.871 0438(41) pJ/𝑐2 = 0.510 998 910(13) MeV/𝑐2 = 0.000 548 579 909 43(23) u gyromagnetic ratio 𝜇𝑒 /𝜇B = −1.001 159 652 1811(7) muon 𝜇 0.188 353 130(11) yg 2.197 03(4) μs 𝐽 = 12 −100000 1, 0 99 % 𝑒− 𝜈𝑒̄ 𝜈𝜇 = 105.658 3668(38) MeV/𝑐2 = 0.113 428 9256(29) u gyromagnetic ratio 𝜇𝜇 /(𝑒ℏ/2𝑚𝜇 ) = −1.001 165 9208(6) tau 𝜏 1.776 84(17) GeV/𝑐2 290.6(1.0) fs 𝐽 = 12 −100000 1, 0 1 el. neutrino < 2 eV/𝑐2 𝐽= 2 1, 0 𝜈e 1 muon < 2 eV/𝑐2 𝐽= 2 1, 0 neutrino 𝜈𝜇 1 tau neutrino < 2 eV/𝑐2 𝐽= 2 1, 0 𝜈𝜏 Elementary matter (fermions): quarks 𝑓 + up 𝑢 1.5 to 3.3 MeV/𝑐2 see proton 𝐼(𝐽𝑃 ) = 12 ( 12 ) + 23 + 12 0000 0, 13 + down 𝑑 3.5 to 6 MeV/𝑐2 see proton 𝐼(𝐽𝑃 ) = 12 ( 12 ) − 13 − 12 0000 0, 13 + strange 𝑠 70 to 130 MeV/𝑐2 𝐼(𝐽𝑃 ) = 0( 12 ) − 13 0−1000 0, 13 + charm 𝑐 1.27(11) GeV/𝑐2 𝐼(𝐽𝑃 ) = 0( 12 ) + 23 00+100 0, 13 6 rotations and statistics – visualizing spin 129 TA B L E 5 (Continued) Elementary particle properties. Particle Mass 𝑚 𝑎 Lifetime 𝜏 Isospin 𝐼, Charge, Lepton or energy spin 𝐽, 𝑐 isospin, & width, 𝑏 parity 𝑃, strange- baryon 𝑒 main decay charge ness, 𝑐 num- modes parity 𝐶 charm, bers beauty, 𝑑 𝐿𝐵 topness: 𝑄𝐼𝑆𝐶𝐵𝑇 + bottom 𝑏 4.20(17) GeV/𝑐2 𝜏 = 1.33(11) ps 𝐼(𝐽𝑃 ) = 0( 12 ) − 13 000−10 0, 13 + top 𝑡 171.2(2.1) GeV/𝑐2 𝐼(𝐽𝑃 ) = 0( 12 ) + 23 0000+1 0, 13 Observed elementary boson Higgs boson 126 GeV/𝑐2 𝐽=0 Motion Mountain – The Adventure of Physics Notes: 𝑎. See also the table of SI prefixes on page 206. About the eV/𝑐2 mass unit, see page 210. 𝑏. The energy width Γ of a particle is related to its lifetime 𝜏 by the indeterminacy relation Γ𝜏 = ℏ. There is a difference between the half-life 𝑡1/2 and the lifetime 𝜏 of a particle: they are related by 𝑡1/2 = 𝜏 ln 2, where ln 2 ≈ 0.693 147 18; the half-life is thus shorter than the lifetime. The unified atomic mass unit u is defined as 1/12 of the mass of a carbon 12 atom at rest and in its ground 1 state. One has 1 u = 12 𝑚(12 C) = 1.660 5402(10) yg. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 𝑐. To keep the table short, its header does not explicitly mention colour, the – confusingly named – charge of the strong interactions. It has to be added to the list of basic object properties. Quantum numbers containing the word ‘parity’ are multiplicative; all others are additive. Parity 𝑃 and charge parity 𝐶 are written as + or −. Time parity 𝑇 (not to be confused with topness 𝑇), better called motion inversion parity, is equal to CP in all known particles. The isospin 𝐼 (or 𝐼Z ) appears twice in the table; it is defined only for up and down quarks and their composites, such as the proton and the neutron. In the literature one also sees references to the so-called 𝐺-parity, defined as 𝐺 = (−1)𝐼𝐶 . The table header also does not mention the weak charge of the particles. The details on weak charge 𝑔, or, more precisely, on the weak isospin, a quantum number assigned to all left-handed fermions (and right-handed anti-fermions), but to no right-handed fermion (and no left-handed Vol. V, page 245 antifermion), are given in the section on the weak interactions. 𝑑. ‘Beauty’ is now commonly called bottomness; similarly, ‘truth’ is now commonly called top- ness. The signs of the quantum numbers 𝑆, 𝐼, 𝐶, 𝐵, 𝑇 can be defined in different ways. In the standard assignment shown here, the sign of each of the non-vanishing quantum numbers is given by the sign of the charge of the corresponding quark. 𝑒. If supersymmetry existed, 𝑅-parity would have to be added to this column. 𝑅-parity is a mul- tiplicative quantum number related to the lepton number 𝐿, the baryon number 𝐵 and the spin 𝐽 through the definition 𝑅 = (−1)3𝐵+𝐿+2𝐽 . All particles from the standard model are 𝑅-even, whereas their conjectured supersymmetric partner particles would be 𝑅-odd. However, super- symmetry is now known to be in contrast with experiment. 𝑓. For the precise definition and meaning of quark masses, see page 233 in volume V. 130 6 rotations and statistics – visualizing spin F I G U R E 61 Illustrating an argument showing why rotations by 4π are equivalent to no rotation at all (see text). Motion Mountain – The Adventure of Physics Spin 1/2 and tethered objects A central result of quantum theory is that spin 1/2 is a possibility in nature, even though this value does not appear in everyday life. For a system to have spin 1/2 means that for such a system only a rotation by two turns is equivalent to none at all, while one by one turn is not. No simple systems with this property exist in everyday life, but such systems do exist in microscopic systems: electrons, neutrinos, silver atoms and molecular radicals copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net all have spin 1/2. Table 4 gives a more extensive list. Vol. I, page 49 The mathematician Hermann Weyl used a simple image to explain that the rotation by two turns is equivalent to zero turns, whereas one turns differs. Take two cones, touching each other at their tips as well as along a line, as shown in Figure 61. Hold one cone and roll the other around it. When the rolling cone, after a full turn around the other cone, i.e., around the vertical axis, has come back to the original position, it has rotated by some Challenge 101 e angle. If the cones are wide, as shown on the left, the final rotation angle is small. The limit of extremely wide cones gives no rotation at all. If the cones are very thin, like needles, the moving cone has rotated by (almost) 720 degrees; this situation is like a coin rolling around a second coin of the same size, both lying on a table. The rolling coins rotates by two turns, thus by 720 degrees. Also in this case, the final rotation angle is small. The result for 0 degrees and for 720 degrees is the same. If we imagine the cone angle to vary continuously, this visualization shows that a 0 degree rotation can be continuously Challenge 102 e changed into a 720 degree rotation. In contrast, a 360 degree rotation cannot be ‘undone’ in this way. There are systems in everyday life that behave like spin 1/2, but they are not simple: all such systems are tethered. The most well-known system is the belt. Figure 62 and Fig- ure 63 show that a rotation by 4π of a belt buckle is equivalent to no rotation at all: this is easily achieved by moving the belt around. You may want to repeat the process by your- Challenge 103 e self, using a real belt or a strip of paper, in order to get a feeling for it. The untangling process is often called the belt trick, but also scissor trick, plate trick, string trick, Philip- pine wine dance or Balinese candle dance. It is sometimes incorrectly attributed to Dirac, 6 rotations and statistics – visualizing spin 131 Motion Mountain – The Adventure of Physics F I G U R E 62 Assume that the belt cannot be observed, but the square object can, and that it represents a particle. The animation then shows that such a particle (the square object) can return to the starting position after rotation by 4π (and not after 2π). Such a ‘belted’ particle thus fulﬁls the deﬁning property of a spin 1/2 particle: rotating it by 4π is equivalent to no rotation at all. The belt thus represents the spinor wave function; for example, a 2π rotation leads to a twist; this means a change of the sign of the wave function. A 4π rotation has no inﬂuence on the wave function. You can repeat the trick at home, with a paper strip. The equivalence is shown here with two attached belts, but the trick works with any positive number of belts! (QuickTime ﬁlm © Antonio Martos) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net F I G U R E 63 The belt trick with a simple belt: a double rotation of the belt buckle is equivalent to no rotation. (QuickTime ﬁlm © Greg Egan) because he used it extensively in his lectures. The human body has such a belt built in: the arm. Just take your hand, put an object on it for clarity, such as a cup, and turn the hand and object by 2π by twisting the arm. After a second rotation the whole system will be untangled again, as shown in Figure 64. The trick is even more impressive when many arms are used. You can put your two hands Challenge 104 e (if you chose the correct starting position) under the cup or you can take a friend or two 132 6 rotations and statistics – visualizing spin 𝛼=0 𝛼 = 2π 𝛼 = 4π F I G U R E 64 The human arm as spin 1/2 model. Motion Mountain – The Adventure of Physics rotating the buckle F I G U R E 65 The generalized either by 4π belt trick, modelling the rotation behaviour of a spin 1/2 particle: independently of the number of bands or tubes or simply rearranging or strings attached, the two the bands gives the situations can be transformed other situation into each other, either by rotating the central object by 4π or by keeping the central copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net object ﬁxed and moving the bands around it. who each keep a hand attached to the cup together with you. The belt trick can still be Challenge 105 e performed, and the whole system untangles after two full turns. This leads us to the most general way to show the connection between tethering and spin 1/2. Just glue any number of threads, belts or tubes, say half a metre long, to some object, as shown in Figure 65. (With many such tails, is not appropriate any more to call it a belt buckle.) Each band is supposed to go to spatial infinity and be attached there. Instead of being attached at spatial infinity, we can also imagine the belts attached to a distant, fixed object, like the arms are attached to a human body. If the object, which represents the particle, is rotated by 2π, twists appear in its tails. If the object is rotated by an additional turn, to a total of 4π, all twists and tangles can be made to disappear, without moving or turning the object. You really have to experience this in order to be- lieve it. And the process really works with any number of bands glued to the object. The website www.evl.uic.edu/hypercomplex/html/dirac.html provides a animation showing this process with four attached belts. In short, all these animations show that belt buckles, and in fact all (sufficiently) tethered systems, return to their original state only after rotations by 4π, and not after rotations by 2π only. Tethered objects behave like spin 1/2 particles. In fact, tethered ob- 6 rotations and statistics – visualizing spin 133 F I G U R E 66 Two belt buckles connected by a belt, one way of visualizing two spin 1/2 particles. Challenge 106 e jects, such as belt buckles, are the only systems that reproduce spin 1/2 properties. In the last part of our adventure we will discover the deep underlying reason for the equivalence between spin 1/2 particles and tethered systems. Exploring the symmetries of wave functions, quantum theory shows that rotations require the existence of spin for all quantum particles. An investigation of the wave func- Motion Mountain – The Adventure of Physics tion shows that wave functions of elementary matter particles behave under rotation like tethered objects. For example, a wave function whose tethered equivalent is tangled ac- quires a negative sign. In summary, quantum theory implies the existence of the slightly counter-intuitive spin 1/2 value. In particular, it appears for elementary matter particles. The extension of the belt trick But why do experiments show that all fermions have half-integer spin and that all bosons copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Page 135 have integer spin? In particular, why do electrons obey the Pauli exclusion principle? At first sight, it is not clear what the spin value has to do with the statistical properties of a particle. In fact, there are several ways to show that rotations and statistics are connected. Ref. 86 The first proof, due to Wolfgang Pauli, used the details of quantum field theory and was so complicated that its essential ingredients were hidden. It took several decades to convince Ref. 87 everybody that a further observation about belts was the central part of the proof. Page 120 Starting from the bulge model of quantum particles shown in Figure 60, we can ima- gine a tube connecting two particles, similar to a belt connecting two belt buckles, as shown in Figure 66. The buckles represent the particles. The tube keeps track of their relative orientation. If one particle/buckle is rotated by 2π along any axis, a twist is inser- ted into the belt. As just shown, if the same buckle is rotated by another 2π, bringing the total to 4π, the ensuing double twist can easily be undone without moving or rotating the buckles. Now we look again at Figure 66. If we take the two buckles and simply swap their positions, a twist is introduced into the belt. If we swap them again, the twist will disap- pear. In short, two connected belt buckles return to their original state only after a double exchange, and not after a single exchange. In other words, if we take each buckle to represent a particle and a twist to mean a factor −1, the belt exactly describes the phase behaviour of spin 1/2 wave functions, both under rotation and under exchange. In particular, we see that rotation and exchange behaviour are related. Similarly, also the belt trick itself can be extended to exchange. Take two buckles that 134 6 rotations and statistics – visualizing spin F I G U R E 67 Extended belt models for two spin 1/2 particles. Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net F I G U R E 68 Assume that the belts cannot be observed, but the square objects can, and that they represent particles. We know from above that belted buckles behave as spin 1/2 particles. The animation shows that two such particles return to the original situation if they are switched in position twice (but not once). Such particles thus fulﬁl the deﬁning property of fermions. (For the opposite case, that of bosons, a simple exchange would lead to the identical situation.) You can repeat the trick at home using paper strips. The equivalence is shown here with two belts per particle, but the trick works with any positive number of belts attached to each buckle. This animation is the essential part of the proof that spin 1/2 particles are fermions. This is called the spin–statistics theorem. (QuickTime ﬁlm © Antonio Martos) are connected with many bands or threads, like in Figure 67 or in Figure 68. The band can connect the particles, or go to spatial infinity, or both. An exchange of the two buckles produces quite a messy tangle. But almost incredibly, in all cases, a second exchange leads Challenge 107 e back to the original situation, if the belts are properly rearranged. You might want to test 6 rotations and statistics – visualizing spin 135 yourself that the behaviour is also valid if additional particles are involved, as long as you always exchange the same two particles twice. We conclude that tethered objects behave like fermions under exchange. These ob- servations together form the spin–statistics theorem for spin 1/2 particles: spin and ex- change behaviour are related. Indeed, these almost ‘experimental’ arguments can be put Ref. 88 into exact mathematical language by studying the behaviour of the configuration space of particles. These investigations result in the following statements: ⊳ Objects of spin 1/2 are fermions.* ⊳ Exchange and rotation of spin 1/2 particles are similar processes. In short, objects that behave like spin 1/2 particles under rotations also behave like fer- mions under exchange. And vice versa. The exchange behaviour of particles determines their statistical properties; the rotation behaviour determines their spin. By extending the belt trick to several buckles, each with several belts, we thus visualized the spin–statistics Motion Mountain – The Adventure of Physics theorem for fermions. Note that all these arguments require three dimensions of space, because there are no tangles (or knots) in fewer or more dimensions.** And indeed, spin exists only in three spatial dimensions. The belt trick leads to interesting puzzles. We saw that a spin 1/2 object can be mod- elled by imagining that a belt leading to spatial infinity is attached to it. If we want to model the spin behaviour with attached one-dimensional strings instead of bands, what Challenge 109 s is the minimum number of strings we need? More difficult is the following puzzle: Can the belt trick be performed if the buckle is glued into a mattress, thus with the mattress copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Challenge 110 d acting like ‘infinitely many’ belts? Angels, Pauli ’ s exclusion principle and the hardness of mat ter Why are we able to knock on a door? Why can stones not fly through tree trunks? How does the mountain we are walking on carry us? Why can’t we walk across walls? In clas- sical physics, we avoided this issue, by taking solidity as a defining property of matter. But we cannot do so any more: we have seen that matter consists mainly of low density electron clouds. The quantum of action thus forces us to explain the quantum of matter. The explanation of the impenetrability of matter is so important that it led to a Nobel prize in physics. The interpenetration of bodies is made impossible by Pauli’s exclusion principle among the electrons inside atoms. Pauli’s exclusion principle states: ⊳ Two fermions cannot occupy the same quantum state. * A mathematical observable behaving like a spin 1/2 particle is neither a vector nor a tensor, as you may Challenge 108 e want to check. An additional concept is necessary; such an observable is called a spinor. We will introduce Page 189 it in detail later on. ** Of course, knots and tangles do exist in higher dimensions. Instead of considering knotted one- dimensional lines, one can consider knotted planes or knotted higher-dimensional hyperplanes. For ex- ample, deformable planes can be knotted in four dimensions and deformable 3-spaces in five dimensions. However, the effective dimensions that produce the knot are always three. 136 6 rotations and statistics – visualizing spin All experiments known confirm the statement. Why do electrons and other fermions obey Pauli’s exclusion principle? The answer Ref. 89 can be given with a beautifully simple argument. We know that exchanging two fermions produces a minus sign in the total wave function. Imagine these two fermions being, as a classical physicist would say, located at the same spot, or as a quantum physicist would say, in the same state. If that could be possible, an exchange would change nothing in the system. But an exchange of fermions must produce a minus sign for the total state. Both possibilities – no change at all as well as a minus sign – cannot be realized at the same time. There is only one way out: two fermions must avoid to ever be in the same state. This is Pauli’s exclusion principle. The exclusion principle is the reason that two pieces of matter in everyday life cannot penetrate each other, but have to repel each other. For example, take a bell. A bell would not work if the colliding pieces that produce the sound would interpenetrate. But in any example of two interpenetrating pieces, the electrons from different atoms would have to be at the same spot: they would have to be in the same states. This is impossible. Pauli’s Motion Mountain – The Adventure of Physics exclusion principle forbids interpenetration of matter. Bells only work because of the exclusion principle. Why don’t we fall through the floor, even though gravity pulls us down, but remain standing on its surface? Again, the reason is Pauli’s exclusion principle. Why does the floor itself not fall? It does not fall, because the matter of the Earth cannot interpenetrate and the atoms cannot made to approach each other than a certain minimal distance. In other words, Pauli’s exclusion principle implies that atomic matter cannot be compressed indefinitely. At a certain stage an effective Pauli pressure appears, so that a compression limit ensues. For this reason for example, planets made of atomic matter – or neutron copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net stars made of neutrons, which also have spin 1/2 and thus also obey the exclusion prin- ciple – do not collapse under their own gravity. The exclusion principle is the reason that atoms are extended electron clouds and that different atoms have different sizes. In fact, the exclusion principle forces the electrons in atoms to form shells. When electrons are added around a nucleus and when one shell is filled, a new shell is started. This is the origin of the periodic systems of the elements. The size of any atom is the size of its last shell. Without the exclusion principle, atoms would be as small as a hydrogen atom. In fact, most atoms are considerably larger. The same argument applies to nuclei: their size is given by the last nucleon shell. Without the exclusion principle, nuclei would be as small as a single proton. In fact, they are usually about 100 000 times larger. The exclusion principle also settles an old question: How many angels can dance on the top of a pin? (Note that angels, if at all, must be made of fermions, as you might want to deduce from the information known about them, and that the top of a pin is a Challenge 111 s single point in space.) Both theory and experiment confirm the answer already given by Ref. 90 Thomas Aquinas in the Middle Ages: Only one angel! The fermion exclusion principle could also be called ‘angel exclusion principle’. To stay in the topic, the principle also shows that ghosts cannot be objects, as ghosts are supposed to be able to traverse walls. Let us sum up. Simplifying somewhat, the exclusion principle keeps things around us in shape. Without the exclusion principle, there would be no three-dimensional objects. Only the exclusion principle fixes the diameter of atomic clouds, keeps these clouds from merging, and holds them apart. This repulsion is the origin for the size of soap, planets 6 rotations and statistics – visualizing spin 137 and neutron stars. All shapes of solids and fluids are a direct consequence of the exclusion principle. In other words, when we knock on a table or on a door, we prove experiment- ally that these objects and our hands are made of fermions. So far, we have only considered fermions of spin 1/2. We will not talk much about particles with odd spin of higher value, such as 3/2 or 5/2. Such particles can all be seen Challenge 112 e as being composed of spin 1/2 entities. Can you confirm this? We did not talk about lower spins than 1/2 either. A famous theorem states that a spin Ref. 82 value between 0 and 1/2 is impossible in three dimensions. Smaller spins are impossible because the largest rotation angle that can be distinguished and measured in three di- mensions is 4π. There is no way to measure a larger angle; the quantum of action makes this impossible. Thus there cannot be any spin value between 0 and 1/2 in nature. Is spin a rotation ab ou t an axis? The spin of a particle behaves experimentally like an intrinsic angular momentum, adds up like angular momentum, is conserved as part of angular momentum, is described like Motion Mountain – The Adventure of Physics angular momentum and has a name synonymous with angular momentum. Despite all this, for many decades a strange and false myth was spread in many physics courses and textbooks around the world: “Spin 1/2, despite its name, is not a rotation about an axis.” It is time to finish with this example of incorrect thinking. Electrons do have spin 1/2 and are charged. Electrons and all other charged particles with spin 1/2 do have a magnetic moment.* A magnetic moment is expected for any rotating charge. In other words, spin 1/2 does behave like rotation. However, assuming that a particle consists of a continuous charge distribution in rotational motion gives the wrong value for the magnetic moment. In the early days of the twentieth century, when copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net physicists were still thinking in classical terms, they concluded that charged spin 1/2 particles thus cannot be rotating. This myth has survived through many textbooks. The correct deduction, however, is that the assumption of continuous charge distribution is wrong. Indeed, charge is quantized; nobody expects that elementary charge is continu- ously spread over space, as that would contradict its quantization. The other reason for the false myth is rotation itself. The myth is based on classical thinking and maintains that any rotating object must have integer spin. Since half integer spin is not possible in classical physics, it is argued that such spin is not due to rotation. But let us recall what rotation is. Both the belt trick for spin 1/2 as well as the integer spin case remind us: a rotation of one body around another is a fraction or a multiple of an exchange. What we call a rotating body in everyday life is a body continuously exchanging the positions of its parts – and vice versa. ⊳ Rotation and exchange are the same process. Now, we just found that spin is exchange behaviour. Since rotation is exchange and spin is exchange, it follows that ⊳ Spin is rotation. * This magnetic moment can easily be measured in an experiment; however, not one of the Stern–Gerlach Challenge 113 ny type. Why not? 138 6 rotations and statistics – visualizing spin 𝑡 𝑥 F I G U R E 69 Equivalence of exchange and rotation in space-time. Motion Mountain – The Adventure of Physics Since we deduced spin, like Wigner, from rotation invariance, this conclusion is not a surprise. In addition, the belt model of a spin 1/2 particle tells us that such a particle Page 131 can rotate continuously without any hindrance. Also the magnetic moment then gets its correct value. In short, we are allowed to maintain that spin is rotation about an axis, without any contradiction to observations, even for spin 1/2. In summary, the belt model shows that also spin 1/2 is rotation, as long as we assume Ref. 91 that only the buckle can be observed, not the belt(s), and that elementary charge is not continuously distributed in space.* copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Since permutation properties and spin properties of fermions are so well described by the belt model, we could be led to the conclusion that these properties might really be consequence of such belt-like connections between particles and the outside world. Maybe for some reason we only observe the belt buckles, not the belts themselves. In the final part of this walk we will discover whether this idea is correct. Rotation requires antiparticles The connection between rotation and antiparticles may be the most astonishing con- clusion from the experiments showing the existence of spin. So far, we have seen that rotation requires the existence of spin, that spin appears when relativity is introduced Vol. II, page 72 into quantum theory, and that relativity requires antimatter. Taking these three state- ments together, the conclusion of the title is not surprising any more: rotation requires antiparticles. Interestingly, there is a simple argument making the same point with the belt model, if it is extended from space alone to full space-time. To learn how to think in space-time, let us take a particle and reduce it to two short tails, so that the particle is a short line segment. When moving in a 2+1 dimensional * Obviously, the exact structure of the electron still remains unclear at this point. Any angular momentum 𝑆 is given classically by 𝑆 = Θ𝜔; however, neither the moment of inertia Θ, connected to the rotation radius and electron mass, nor the angular velocity 𝜔 are known at this point. We have to wait quite a while, until the final part of our adventure, to find out more. 6 rotations and statistics – visualizing spin 139 t t t t t x x x x x F I G U R E 70 Belts in space-time: rotation and antiparticles. Challenge 114 ny space-time, the particle is described by a ribbon. Playing around with ribbons in space- time, instead of belts in space, provides many interesting conclusions. For example, Fig- Motion Mountain – The Adventure of Physics ure 69 shows that wrapping a rubber ribbon around the fingers can show, again, that a rotation of a body by 2π in presence of a second one is the same as exchanging the pos- itions of the two bodies.* Both sides of the hand transform the same initial condition, at one edge of the hand, to the same final condition at the other edge. We have thus suc- cessfully extended a known result from space to space-time: rotation and exchange are equivalent. If you think that Figure 69 is not a satisfying explanation, you are right. A more sat- isfying explanation must include a smooth sequence of steps realizing the equivalence between rotation and exchange. This is shown in Figure 70. We assume that each particle copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net is described by a segment; in the figure, the two segments lie horizontally. The leftmost diagram shows two particles: one at rest and one being rotated by 2π. The deformation of the ribbons shows that this process is equivalent to the exchange in position of two particles, which is shown in the rightmost diagram. But the essential point is made by the intermediate diagrams. We note that the se- quence showing the equivalence between rotation and exchange requires the use of a loop. But such a loop in space-time describes the appearance of a particle–antiparticle pair! In other words, without antiparticles, the equivalence of rotation and exchange would not hold. In short, rotation in space-time requires the existence of antiparticles. Why is fencing with laser beams impossible? When a sword is approaching dangerously, we can stop it with a second sword. Many old films use such scenes. When a laser beam is approaching, it is impossible to fend it off with a second beam, despite all science fiction films showing so. Banging two laser beams against each other is impossible. The above explanation of the spin–statistics theorem shows why. The electrons in the swords are fermions and obey the Pauli exclusion principle. Fer- mions make matter impenetrable. On the other hand, the photons in laser beams are * Obviously, the full argument would need to check the full spin 1/2 model of Figure 65 in four-dimensional Challenge 115 ny space-time. But doing this is not an easy task; there is no good visualization yet. 140 6 rotations and statistics – visualizing spin J=0 J = 1/2 J=1 F I G U R E 71 Some visualizations of spin representations. Motion Mountain – The Adventure of Physics bosons. Two bosons can be in the same state; bosons allow interpenetration. Matter is impenetrable because at the fundamental level it is composed of fermions. Radiation is composed of bosons; light beams can cross each other. The distinction between fermi- ons and bosons thus explains why objects can be touched while images cannot. In the Vol. I, page 98 first part of our mountain ascent we started by noting this difference; now we know its origin. Spin, statistics and composition copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Under rotations, integer spin particles behave differently from half-integer particles. In- teger spin particles do not show the strange sign changes under rotations by 2π. In the belt imagery, integer spin particles need no attached strings. In particular, a spin 0 particle obviously corresponds to a sphere. Models for other important spin values are shown in Figure 71. Exploring their properties in the same way as above, we arrive at the full spin–statistics theorem: ⊳ Exchange and rotation of objects are similar processes. ⊳ Objects of half-integer spin are fermions. They obey the Pauli exclusion principle. ⊳ Objects of integer spin are bosons. Challenge 116 e You might prove by yourself that this suffices to show the following rule: ⊳ Composites of bosons, as well as composites of an even number of fermi- ons (at low energy), are bosons; composites of an uneven number of fermi- ons are fermions.* Challenge 117 s * This rule implies that spin 1 and higher can also be achieved with tails; can you find such a representation? Note that composite fermions can be bosons only up to that energy at which the composition breaks down. Otherwise, by packing fermions into bosons, we could have fermions in the same state. 6 rotations and statistics – visualizing spin 141 These connections express basic characteristics of the three-dimensional world in which we live. To which class of particles do tennis balls, people, trees, mountains and all other Challenge 118 s macroscopic objects belong? The size and densit y of mat ter The three spatial dimensions have many consequences for physical systems. We know that all matter is made of fermions, such as electrons, protons and neutrons. The exclu- sion principle has an interesting consequence for systems made of 𝑁 identical fermions; such systems obey the following expression for momentum 𝑝 and size 𝑙: Δ𝑝 Δ𝑙 ≳ 𝑁1/3 ℏ . (67) Challenge 119 e Can you derive it? This extended indeterminacy relation provides a simple way to estimate Ref. 92 the spatial size of matter systems. In particular, the extended indeterminacy relation im- Motion Mountain – The Adventure of Physics plies that the average energy per quanton increases with quanton density. Can you show Challenge 120 e this? The extended indeterminacy relation implies that matter systems whose extension is due to electrons – thus all condensed matter systems – essentially have similar matter and energy densities. The extended indeterminacy relation also implies that nuclei, which are composed of protons and neutrons, all have essentially the same matter density. For bosons, the components of radiation, there is no extended indeterminacy relation, as the number of components 𝑁 in a particular quantum state does not have any effect or limits. The indeterminacy relation thus does not limit the power density of laser light; and indeed, the power density of laser beams varies much more than the matter density copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net of solids. The indeterminacy relation highlights a further difference between matter and radi- Page 48 ation. As we saw above, a system of 𝑁 identical bosons, such as a laser beam, obeys an indeterminacy between the number and the phase which is easily derived from the energy–time indeterminacy relation. The number–phase relation can be written, approx- imately, as Δ𝑁 Δ𝜑 ≳ 1 . (68) It is important in the use of lasers in precision experiments. The relation limits how close a system can get to a pure sine wave; indeed for a pure sine wave, the indeterminacy product would be zero. For fermions, where the maximum number in the same state is 1, the number–phase uncertainty relation reduces to a total uncertainty on the phase. In other words, we find Page 101 – again – that we cannot have fermion beams that behave as waves. There are no classical fermion waves, no coherent fermion beams, in nature. A summary on spin and indistinguishabilit y The quantum of action ℏ implies that physical systems are made of two types of indistin- guishable quantum particles: bosons and fermions. The two possible exchange behaviours are related to the particle spin value, because exchange is related to rotation. The connec- 142 6 rotations and statistics – visualizing spin tion between spin and rotation implies that antiparticles exist. It also implies that spin is intrinsically a three-dimensional phenomenon. Experiments show that radiation is made of elementary particles that behave as bo- sons. Bosons have integer spin. Two or more bosons, such as two photons, can share the same state. This sharing makes laser light possible. Experiments show that matter is made of elementary particles that behave as fermi- ons. Fermions have half-integer spin. They obey Pauli’s exclusion principle: two fermi- ons cannot be in the same state. The exclusion principle between electrons explains the structure and (partly) the size of atoms, as well as the chemical behaviour of atoms, as we will find out later on. Together with the electrostatic repulsion of electrons, the exclusion principle explains the incompressibility of matter and its lack of impenetrability. Fermions make matter ‘hard’, bosons allow light beams to cross. Limits and open questions of quantum statistics The topic of quantum particle statistics remains a research field in theoretical and ex- Motion Mountain – The Adventure of Physics perimental physics. In particular, researchers have searched and still are searching for generalizations of the possible exchange behaviours of particles. In two spatial dimensions, the effect of a particle exchange on the wave function is Page 137 a continuous phase, in contrast to three dimensions, where the result is a sign. Two- dimensional quantum objects are therefore called anyons because they can have ‘any’ spin. Anyons appear as quasi-particles in various experiments in solid state physics, be- cause the set-up is often effectively two-dimensional. The fractional quantum Hall effect, perhaps the most interesting discovery of modern experimental physics, has pushed any- Vol. V, page 107 ons onto the stage of modern research. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Other theorists generalized the concept of fermions in other ways, introducing par- Ref. 93 afermions, parabosons, plektons and other hypothetical concepts. Oscar Greenberg has spent most of his professional life on this issue. His conclusion is: ⊳ In 3 + 1 space-time dimensions, only fermions and bosons exist. Can you show that this result implies that the ghosts appearing in Scottish tales do not Challenge 121 s exist? From a different viewpoint, the belt model of spin 1/2 invites to study the behaviour of braids, open links and knots. (In mathematics, braids and open links are made of strands extending to infinity.) This fascinating part of mathematical physics has become import- ant with in modern unified theories, which all state that particles, especially at high en- ergies, are not point-like, but extended entities. The quest is to understand what happens to permutation symmetry in a unified theory of nature. A glimpse of the difficulties ap- pears already above: how can Figures 60, 65 and 70 be reconciled and combined? We will Vol. VI, page 174 settle this issue in the final part of our mountain ascent. Chapter 7 SU PE R P O SI T ION S A N D PROBA BI L I T I E S – QUA N T UM T H E ORY W I T HOU T I DE OLO G Y “ The fact that an adequate philosophical presentation has been so long delayed is no doubt caused by the fact that Niels Bohr Ref. 94 brainwashed a whole generation of theorists into thinking that the job was done fifty years Motion Mountain – The Adventure of Physics ” ago. Murray Gell-Mann W hy is this famous physical issue arousing such strong emotions? In particular, ho is brainwashed, Gell-Mann, the discoverer of the quarks, or most of the orld’s physicists working on quantum theory who follow Niels Bohr’s opinion? In the twentieth century, quantum mechanics has thrown many in disarray. We have a simple aim: we want to understand quantum theory. Quantum mechanics is unfamiliar for two reasons: it allows superpositions and it leads to probabilities. In this chapter we explore and clarify these two topics – until we understand quantum theory. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Probabilities appear whenever an aspect of a microscopic system is measured. The quantum of action, the smallest change value found in nature, leads to the appearance of probabilities in measurements. Superpositions appear because the quantum of action radically changed the two most basic concepts of classical physics: state and system. The state is not defined and de- scribed any more by the specific values taken by position and momentum, but by the specific wave function ‘taken’ by the position and momentum operators.** In addi- tion, in classical physics a system was described and defined as a set of permanent as- pects of nature; permanence was defined as negligible interaction with the environment. Quantum mechanics shows that these definitions have to be modified. Clarifying the origin of superpositions and probabilities, as well as the concepts of sys- tem and state, will help us to avoid getting lost on our way to the top of Motion Moun- tain. Indeed, quite a number of researchers have lost their way since quantum theory appeared, including important physicists like Murray Gell-Mann and Steven Weinberg. ** It is equivalent, and often conceptually clearer, to say that the state is described by a complete set of commuting operators. In fact, the discussion of states is somewhat simplified in the Heisenberg picture. However, here we study the issue in the Schrödinger picture only, i.e., using wave functions. 144 7 superpositions and probabilities F I G U R E 72 An artist’s Every such `artistic impression’ is wrong. impression of a macroscopic Challenge 122 s (Why?) superposition is impossible – because such superpositions are not found in our environment. Why are people either dead or alive? The evolution equation of quantum mechanics is linear in the wave function; the linearity reflects the existence of superpositions. Superpositions imply that we can imagine and try to construct systems where the state 𝜓 is a superposition of two radically distinct situations, such as those of a dead and of a living cat. This famous fictional animal is Motion Mountain – The Adventure of Physics called Schrödinger’s cat after the originator of the example. Is it possible to produce it? And how would it evolve in time? We can ask the same two questions in other situations. For example, can we produce a superposition of a state where a car is inside a closed garage with a state where the car is outside? What happens then? Macroscopic superpositions are strange. Such situations are not observed in everyday life, and only very rarely in the laboratory. The reason for this rareness is an important aspect of what is often called the ‘interpretation’ of quantum mechanics. In fact, such strange situations are possible, and the superposition of macroscopically distinct states has actually been observed in a few cases, though not for cats, people or cars. To get an idea of the constraints, let us specify the situation in more detail.* copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Macroscopic superpositions, coherence and incoherence The object of discussion are linear superpositions of the type 𝜓 = 𝑎𝜓𝑎 + 𝑏𝜓𝑏 , where 𝜓𝑎 and 𝜓𝑏 are macroscopically distinct states of the system under discussion, and where 𝑎 and 𝑏 are some complex coefficients. States are called macroscopically distinct when each state corresponds to a different macroscopic situation, i.e., when the two states can be distinguished using the concepts or measurement methods of classical physics. In par- ticular, this means that the physical action necessary to transform one state into the other must be much larger than ℏ. For example, two different positions of a body composed of a large number of atoms are macroscopically distinct. The state of a cat that is living and of the same cat when it is dead also differ by many quanta of action. A ‘strange’ situation is thus a superposition of macroscopically distinct states. Let us work out the essence of such macroscopic superpositions more clearly. Given two macroscopically distinct states 𝜓𝑎 and 𝜓𝑏 , any superposition of the type 𝜓 = 𝑎𝜓𝑎 + 𝑏𝜓𝑏 is called a pure state. Since the states 𝜓𝑎 and 𝜓𝑏 can interfere, one also talks about a (phase) coherent superposition. In the case of a superposition of macroscopically distinct states, * Most what can be said about this topic has been said by three important researchers: Niels Bohr, one of the fathers of quantum physics, John von Neumann, who in the nineteen-thirties stressed the differences Ref. 95 between evolution and decoherence, and by Heinz Dieter Zeh, who in the nineteen-seventies stressed the Ref. 96 importance of baths and the environment in the decoherence process. 7 quantum theory without ideology 145 the scalar product 𝜓𝑎† 𝜓𝑏 is obviously vanishing. In case of a coherent superposition, the coefficient product 𝑎∗ 𝑏 is different from zero. This fact can also be expressed with the help of the density matrix 𝜌 of the system, defined as 𝜌 = 𝜓 ⊗ 𝜓† . In the present case it is given by 𝜌pure = 𝜓 ⊗ 𝜓† = |𝑎|2 𝜓𝑎 ⊗ 𝜓𝑎† + |𝑏|2 𝜓𝑏 ⊗ 𝜓𝑏† + 𝑎 𝑏∗ 𝜓𝑎 ⊗ 𝜓𝑏† + 𝑎∗ 𝑏 𝜓𝑏 ⊗ 𝜓𝑎† |𝑎|2 𝑎 𝑏∗ 𝜓𝑎† = (𝜓𝑎 , 𝜓𝑏 ) ( ∗ ) ( †) . (69) 𝑎 𝑏 |𝑏|2 𝜓𝑏 We can then say that whenever the system is in a pure, or coherent state, then its density matrix, or density functional, contains off-diagonal terms of the same order of magnitude as the diagonal ones.* Such a density matrix corresponds to the above-mentioned strange situations that we never observe in daily life. We will shortly understand why. We now have a look at the opposite situation, a density matrix for macroscopic distinct states with vanishing off-diagonal elements. For two states, the example Motion Mountain – The Adventure of Physics 𝜌mixed = |𝑎|2 𝜓𝑎 ⊗ 𝜓𝑎† + |𝑏|2 𝜓𝑏 ⊗ 𝜓𝑏† |𝑎|2 0 𝜓𝑎† = (𝜓𝑎 , 𝜓𝑏 ) ( ) ( ) (71) 0 |𝑏|2 𝜓𝑏† describes a system which possesses no phase coherence at all. (Here, ⊗ denotes the non- commutative dyadic product or tensor product which produces a tensor or matrix start- ing from two vectors.) Such a diagonal density matrix cannot be that of a pure state; the density matrix describes a system which is in the state 𝜓𝑎 with probability |𝑎|2 and copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net which is in the state 𝜓𝑏 with probability |𝑏|2 . Such a system is said to be in a mixed state, because its state is not known, or equivalently, is in a (phase) incoherent superposition: in- terference effects cannot be observed in such a situation. A system described by a mixed state is always either in the state 𝜓𝑎 or in the state 𝜓𝑏 . In other words, a diagonal dens- ity matrix for macroscopically distinct states is not in contrast, but in agreement with everyday experience. In the picture of density matrices, the non-diagonal elements contain the difference between normal, i.e., incoherent, and unusual or strange, i.e., coherent, superpositions. The experimental situation is clear: for macroscopically distinct states, only diagonal density matrices are observed in everyday life. Almost all systems in a coherent macro- scopic superposition somehow lose their off-diagonal matrix elements. How does this process of decoherence – also called disentanglement in certain settings – take place? The density matrix itself shows the way. * Using the density matrix, we can rewrite the evolution equation of a quantum system: d𝜌 𝑖 𝜓̇ = −𝑖𝐻𝜓 becomes = − [𝐻, 𝜌] . (70) d𝑡 ℏ Both are completely equivalent. (The new expression is sometimes also called the von Neumann equation.) We won’t actually do any calculations here. The expressions are given so that you recognize them when you encounter them elsewhere. 146 7 superpositions and probabilities Decoherence is due to baths Ref. 97 In thermodynamics, the density matrix 𝜌 for a large system is used for the definition of Challenge 123 ny its entropy 𝑆 – and of all its other thermodynamic quantities. These studies show that 𝑆 = −𝑘 tr (𝜌 ln 𝜌) (72) where tr denotes the trace, i.e., the sum of all diagonal elements, and 𝑘 is the Boltzmann constant. The expression is thus the quantum mechanical definition of entropy. We now remind ourselves that a physical system with a large and constant entropy is called a bath. In simple physical terms, a bath is a system to which we can ascribe a temperature. More precisely, A (physical) bath – also called a (thermodynamic) reservoir – is any large system for which the concept of equilibrium can be applied. Motion Mountain – The Adventure of Physics Experiments show that in practice, this is equivalent to the condition that a bath consists of many interacting subsystems. For this reason, all macroscopic quantities describing the state of a bath show small, irregular fluctuations, a property that will be of central importance shortly. An everyday bath is also a physical bath: indeed, a thermodynamic bath is similar to an extremely large warm water bath, one for which the temperature does not change even if we add some cold or warm water to it. The physical concept of bath, or reservoir, is thus an abstraction and a generalization of the everyday concept of bath. Other examples of physical baths are: an intense magnetic field, a large amount of gas, or a large solid. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net (The meanings of ‘intense’ and ‘large’ of course depend on the system under study.) The definition (72) of entropy tells us that the loss of off-diagonal elements corres- Challenge 124 s ponds to an increase in entropy. In addition, any increase in entropy of a reversible sys- tem, such as the quantum mechanical system in question, is due to an interaction with a bath. In short, decoherence is due to interaction with a bath. In addition, decoherence is a process that increases entropy: decoherence is irreversible. We will now show that baths are everywhere, that decoherence thus takes place everywhere and all the time, and that therefore, macroscopic superpositions are (almost) never observed. How baths lead to decoherence – scat tering Where is the bath interacting with a typical system? The bath must be outside the system we are talking about, i.e., in its environment. Indeed, we know experimentally that a typ- ical environment is large and characterized by a temperature. Some examples are listed in Table 6. In short, ⊳ Any environment is a bath. We can even go further: for every experimental situation, there is a bath interacting with the system under study. Indeed, every system which can be observed is not isolated, as it obviously interacts at least with the observer; and every observer by definition contains 7 quantum theory without ideology 147 TA B L E 6 Common and less common baths with their main properties. B at h t y p e T e m p e r - Wa v e - Pa r - Cross Hit time at u r e length ticle s e c t i o n 1/𝜎𝜑 f o r flux ( at o m ) 𝑇 𝜆 eff 𝜑 𝜎 a t o m𝑎 b a l l𝑎 matter baths solid, liquid 300 K 10 pm 1031 /m2 s 10−19 m2 10−12 s 10−25 s air 300 K 10 pm 1028 /m2 s 10−19 m2 10−9 s 10−22 s laboratory vacuum 50 mK 10 μm 1018 /m2 s 10−19 m2 10 s 10−12 s photon baths sunlight 5800 K 900 nm 1023 /m2 s 10−4 s 10−17 s ‘darkness’ 300 K 20 μm 1021 /m2 s 10−2 s 10−15 s cosmic microwaves 2.7 K 2 mm 1017 /m2 s 102 s 10−11 s Motion Mountain – The Adventure of Physics terrestrial radio waves Casimir effect very large Unruh radiation of Earth 40 zK very large nuclear radiation baths radioactivity 10 f m 1 /m2 s 10−25 m2 1025 s 1012 s cosmic radiation >1000 K 10 f m 10−2 /m2 s 10−25 m2 1027 s 1014 s solar neutrinos ≈ 10 MK 10 f m 1011 /m2 s 10−47 m2 1036 s 1015 s cosmic neutrinos 2.0 K 3 mm 1017 /m2 s 10−62 m2 1045 s 1024 s copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net gravitational baths gravitational radiation 5 ⋅ 1031 K 10−35 m very large 𝑎. Values are rough estimates. The macroscopic ball is assumed to have a 1 mm size. a bath, as we will show in more detail shortly. Usually however, the most important baths we have to take into consideration are the atmosphere around a system, the radiation or electromagnetic fields interacting with the system, or, if the system itself is large enough to have a temperature, those degrees of freedom of the system which are not involved in the superposition under investigation. Since every physical system is in contact with a bath, every density matrix of a macro- scopic superposition will lose its diagonal elements eventually. At first sight, this direc- tion of thought is not convincing. The interactions of a system with its environment can be made extremely small by using clever experimental set-ups; that would imply that the time for decoherence can be made extremely large. Thus we need to check how much time a superposition of states needs to decohere. It turns out that there are two standard ways to estimate the decoherence time: either by modelling the bath as large number of colliding particles, or by modelling it as a continuous field. If the bath is described as a set of particles randomly hitting the microscopic system, it is best characterized by the effective wavelength 𝜆 eff of the particles and by the average Challenge 125 ny interval 𝑡hit between two hits. A straightforward calculation shows that the decoherence 148 7 superpositions and probabilities time 𝑡𝑑 is in any case smaller than this time interval, so that 1 𝑡𝑑 ⩽ 𝑡hit = , (73) 𝜑𝜎 where 𝜑 is the flux of particles and 𝜎 the cross-section for the hit.* Typical values are given in Table 6. We easily note that for macroscopic objects, decoherence times are ex- tremely short. (We also note that nuclear and gravitational effects lead to large decoher- ence times and thus can be neglected.) Scattering leads to fast decoherence of macroscopic systems. However, for atoms or smaller systems, the situation is different, as expected. Microscopic systems can show long decoherence times. We note that the quantum of action ℏ appears in the expression for the decoherence time, as it appears in the area 𝜎. Decoherence is a quantum process. How baths lead to decoherence – relaxation Motion Mountain – The Adventure of Physics A second method to estimate the decoherence time is also common. Any interaction of a system with a bath is described by a relaxation time 𝑡𝑟 . The term relaxation designates any process which leads to the return to the equilibrium state. The terms damping and friction are also used. In the present case, the relaxation time describes the return to equi- librium of the combination bath and system. Relaxation is an example of an irreversible evolution. A process is called irreversible if the reversed process, in which every com- ponent moves in opposite direction, is of very low probability.** For example, it is usual that a glass of wine poured into a bowl of water colours the whole water; it is very rarely observed that the wine and the water separate again, since the probability of all water copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net and wine molecules to change directions together at the same time is rather low, a state of affairs making the happiness of wine producers and the despair of wine consumers. Now let us simplify the description of the bath. We approximate it by a single, un- specified, scalar field which interacts with the quantum system. Due to the continuity of space, such a field has an infinity of degrees of freedom. They are taken to model the many degrees of freedom of the bath. The field is assumed to be in an initial state where its degrees of freedom are excited in a way described by a temperature 𝑇. The interac- tion of the system with the bath, which is at the origin of the relaxation process, can be described by the repeated transfer of small amounts of energy 𝐸hit until the relaxation * The decoherence time is derived by studying the evolution of the density matrix 𝜌(𝑥, 𝑥 ) of objects local- 2 ized at two points 𝑥 and 𝑥 . One finds that the off-diagonal elements follow 𝜌(𝑥, 𝑥 , 𝑡) = 𝜌(𝑥, 𝑥 , 0)e−Λ𝑡(𝑥−𝑥 ) , where the localization rate Λ is given by Λ = 𝑘2 𝜑𝜎eff (74) where 𝑘 is the wave number, 𝜑 the flux and 𝜎eff the cross-section of the collisions, i.e., usually the size of the Ref. 98 macroscopic object. One also finds the surprising result that a system hit by a particle of energy 𝐸hit collapses the density Ref. 99 matrix roughly down to the de Broglie (or thermal de Broglie) wavelength of the hitting particle. Both results together give the formula above. ** Beware of other definitions which try to make something deeper out of the concept of irreversibility, such as claims that ‘irreversible’ means that the reversed process is not at all possible. Many so-called ‘contradictions’ between the irreversibility of processes and the reversibility of evolution equations are due to this mistaken interpretation of the term ‘irreversible’. 7 quantum theory without ideology 149 process is completed. The objects of interest in this discussion, like the mentioned cat, person or car, are described by a mass 𝑚. Their main characteristic is the maximum energy 𝐸𝑟 which can be transferred from the system to the environment. This energy describes the interac- tions between system and environment. The superpositions of macroscopic states we are interested in are solutions of the Hamiltonian evolution of these systems. The initial coherence of the superposition, so disturbingly in contrast with our every- Ref. 100 day experience, disappears exponentially within a decoherence time 𝑡𝑑 given by* 𝐸hit e𝐸hit /𝑘𝑇 − 1 𝑡𝑑 = 𝑡𝑟 (77) 𝐸𝑟 e𝐸hit /𝑘𝑇 + 1 where 𝑘 is again the Boltzmann constant and like above, 𝐸𝑟 is the maximum energy which can be transferred from the system to the environment. Note that one always has 𝑡𝑑 ⩽ 𝑡𝑟 . After the decoherence time 𝑡𝑑 is elapsed, the system has evolved from the coherent to Motion Mountain – The Adventure of Physics the incoherent superposition of states, or, in other words, the density matrix has lost its off-diagonal terms. One also says that the phase coherence of this system has been des- troyed. Thus, after a time 𝑡𝑑 , the system is found either in the state 𝜓𝑎 or in the state 𝜓𝑏 , respectively with the probability |𝑎|2 or |𝑏|2 , and not any more in a coherent superpos- ition which is so much in contradiction with our daily experience. Which final state is selected depends on the precise state of the bath, whose details were eliminated from the calculation by taking an average over the states of its microscopic constituents. The important result is that for all macroscopic objects, the decoherence time 𝑡𝑑 is ex- tremely small. In order to see this more clearly, we can study a special simplified case. A copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net macroscopic object of mass 𝑚, like the mentioned cat or car, is assumed to be at the same time in two locations separated by a distance 𝑙, i.e., in a superposition of the two corres- ponding states. We further assume that the superposition is due to the object moving as a quantum mechanical oscillator with frequency 𝜔 between the two locations; this is the simplest possible system that shows superpositions of an object located in two different positions. The energy of the object is then given by 𝐸𝑟 = 𝑚𝜔2 𝑙2 , and the smallest transfer energy 𝐸hit = ℏ𝜔 is the difference between the oscillator levels. In a macroscopic situ- ation, this last energy is much smaller than 𝑘𝑇, so that from the preceding expression we Ref. 102 get 𝐸2hit ℏ2 𝜆2𝑇 𝑡𝑑 = 𝑡𝑟 = 𝑡𝑟 = 𝑡𝑟 2 (78) 2𝐸𝑟 𝑘𝑇 2𝑚𝑘𝑇𝑙2 𝑙 * This result is derived as in the above case. A system interacting with a bath always has an evolution given Ref. 101 by the general form d𝜌 𝑖 1 = − [𝐻, 𝜌] − ∑[𝑉 𝜌, 𝑉𝑗† ] + [𝑉𝑗 , 𝜌𝑉𝑗† ] , (75) d𝑡 ℏ 2𝑡𝑜 𝑗 𝑗 where 𝜌 is the density matrix, 𝐻 the Hamiltonian, 𝑉 the interaction, and 𝑡𝑜 the characteristic time of the Challenge 126 ny interaction. Are you able to see why? Solving this equation, one finds for the elements far from the diagonal 𝜌(𝑡) = 𝜌0 e−𝑡/𝑡0 . In other words, they disappear with a characteristic time 𝑡𝑜 . In most situations one has a relation of the form 𝐸 𝑡0 = 𝑡𝑟 hit = 𝑡hit (76) 𝐸𝑟 or some variations of it, as in the example above. 150 7 superpositions and probabilities in which the frequency 𝜔 has disappeared. The quantity 𝜆 𝑇 = ℏ/√2𝑚𝑘𝑇 is called the thermal de Broglie wavelength of a particle. We note again that the quantum of action ℏ appears in the expression for the deco- herence time. Decoherence is a quantum process. It is straightforward to see that for practically all macroscopic objects the typical deco- herence time 𝑡𝑑 is extremely short. For example, setting 𝑚 = 1 g, 𝑙 = 1 mm and 𝑇 = 300 K we get 𝑡𝑑 /𝑡𝑟 = 1.3⋅10−39 . Even if the interaction between the system and the environment would be so weak that the system would have as relaxation time the age of the universe, which is about 4 ⋅ 1017 s, the time 𝑡𝑑 would still be shorter than 5 ⋅ 10−22 s, which is over a million times faster than the oscillation time of a beam of light (about 2 fs for green light). For Schrödinger’s cat, the decoherence time would be even shorter. These times are so short that we cannot even hope to prepare the initial coherent superposition, let alone to observe its decay or to measure its lifetime. For microscopic systems however, the situation is different. For example, for an elec- tron in a solid cooled to liquid helium temperature we have 𝑚 = 9.1 ⋅ 10−31 kg, and typ- Motion Mountain – The Adventure of Physics ically 𝑙 = 1 nm and 𝑇 = 4 K; we then get 𝑡𝑑 ≈ 𝑡𝑟 and therefore the system can stay in a coherent superposition until it is relaxed, which confirms that for this case coherent effects can indeed be observed if the system is kept isolated. A typical example is the be- Ref. 103 haviour of electrons in superconducting materials. We will mention a few more below. In 1996 the first actual measurement of decoherence times was published by the Paris Ref. 104 team led by Serge Haroche. It confirmed the relation between the decoherence time and the relaxation time, thus showing that the two processes have to be distinguished at mi- croscopic scale. In the meantime, many other experiments confirmed the decoherence Ref. 105 process with its evolution equation, both for small and large values of 𝑡𝑑 /𝑡𝑟 . A particularly copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Ref. 106 beautiful experiment has been performed in 2004, where the disappearance of two-slit interference for 𝐶70 molecules was observed when a bath interacts with them. Summary on decoherence, life and death Our exploration showed that decoherence results from coupling to a bath in the environ- ment. Decoherence is a quantum statistical effect, i.e., a thermodynamic effect. Decoher- ence follows from quantum theory, is an irreversible process, and thus occurs automat- ically. Above all, decoherence is a process that has been observed in experiments. The estimates of decoherence times in everyday life told us that both the preparation and the survival of superpositions of macroscopically different states is made impossible by the interaction with any bath found in the environment. This is the case even if the usual measure of this interaction, given by the friction of the motion of the system, is very small. Even if a macroscopic system is subject to an extremely low friction, leading to a very long relaxation time, its decoherence time is still vanishingly short. Only care- fully designed microscopic systems in expensive laboratory set-ups can reach substantial decoherence times. Our everyday environment is full of baths. Therefore, ⊳ Coherent superpositions of macroscopically distinct states never appear in everyday life, due to the rapid decoherence times induced by baths in the environment. 7 quantum theory without ideology 151 Cars cannot be in and out of a garage at the same time. We cannot be dead and alive at the same time. An illustration of a macroscopic superposition – see Figure 122 – is impossible. In agreement with the explanation, coherent superpositions of macroscopic Page 155 states appear in some special laboratory situations. What is a system? What is an object? In classical physics, a system is a part of nature that can be isolated from its environment. However, quantum mechanics tells us that isolated systems do not exist, since interac- tions cannot be made vanishingly small. The contradiction can be solved with the results above: they allow us to define the concept of system with more accuracy. ⊳ A system is any part of nature that interacts incoherently with its environ- ment. This implies: Motion Mountain – The Adventure of Physics ⊳ An object is a part of nature interacting with its environment only through baths. In particular, we get: ⊳ A system is called microscopic or quantum mechanical and can described by a wave function 𝜓 whenever — it is almost isolated, with 𝑡evol = ℏ/Δ𝐸 < 𝑡r , and copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Ref. 107 — it is in incoherent interaction with its environment. In short, a microscopic or quantum mechanical system can be described by a wave func- tion only if it interacts incoherently and weakly with its environment. (For such a system, the energy indeterminacy Δ𝐸 is larger than the relaxation energy.) In contrast, a bath is never isolated in the sense just given, because the evolution time of a bath, the time scale during which its properties change, is always much larger than its relaxation time. Since all macroscopic bodies are in contact with baths – or even contain one – they cannot be described by a wave function. In particular, it is impossible to describe any measuring apparatus with the help of a wave function. We thus conclude: ⊳ A macroscopic system is a system with a decoherence time much shorter than any other evolution time of its constituents. Obviously, macroscopic systems also interact incoherently with their environment. Thus cats, cars and television news speakers are all macroscopic systems. Entanglement One possibility is left over by the two definitions of system: what happens in the situation in which the interactions with the environment are coherent? We will encounter some ex- 152 7 superpositions and probabilities amples shortly. Following the definitions, they are neither microscopic nor macroscopic systems. ⊳ A ‘system’ in which the interaction with its environment is coherent is called entangled. Such ‘systems’ are not described by a wave function, and strictly speaking, they are not systems. In these situations, when the interaction is coherent, one speaks of entangle- ment. For example, one says that a particle or set of particles is said to be entangled with its environment. Entangled, i.e., coherently interacting systems can be divided, but must be disen- tangled when doing so. The act of division leads to detached entities; detached entit- ies interact incoherently. Quantum theory shows that nature is not made of detached Motion Mountain – The Adventure of Physics entities, but that it is made of detachable entities. In quantum theory, the criterion of detachment is the incoherence of interaction. Coherent superpositions imply the sur- prising consequence that there are systems which, even though they look being made of detached parts, are not. Entanglement poses a limit to detachment. All surprising prop- erties of quantum mechanics, such as Schrödinger’s cat, are consequences of the classical prejudice that a system made of two or more parts can obviously be detached into two subsystems without disturbance. But coherent superpositions, or entangled systems, do not allow detachment without disturbance. Whenever we assume to be able to detach entangled systems, we get strange or incorrect conclusions, such as apparent faster-than- light propagation, or, as one says today, non-local behaviour. Let us have a look at a few copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net typical examples. Entangled situations are observed in many experiments. For example, when an elec- tron and a positron annihilate into two photons, the polarisations of these two photons are entangled, as measured already in 1949. Also when an excited atom decays in steps, emitting two photons, the photon polarisations are entangled, as was first shown in 1966 with the help of calcium atoms. Similarly, when an unstable molecule in a singlet state, i.e., in a spin 0 state, decays or splits into debris, the spins of the debris are entangled, as observed in the 1970s. Also the spontaneous parametric down-conversion of photons produces entanglement. In a non-linear optical material, an incoming photon is conver- ted into two outgoing photons whose added energies correspond to the energy of the incoming photon. In this case, the two outgoing photons are entangled both in their po- larisation and in their direction. In 2001, the spins of two extremely cold caesium gas samples, with millions of atoms each and located a few millimetres apart, have been en- tangled. Also position entanglement has been regularly observed, for example for closely spaced ions inside ion traps. Is quantum theory non-lo cal? A bit ab ou t the Einstein–Podolsky–Rosen parad ox “ [Mr. Duffy] lived a little distance away from his ” body ... James Joyce, A Painful Case 7 quantum theory without ideology 153 space collapse t1 t2 Motion Mountain – The Adventure of Physics t3 t4 F I G U R E 73 Quantum mechanical slit screen space motion: an electron wave function (actually its module squared) from the moment it passes a slit until it hits a screen. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net It is often suggested, incorrectly, that wave function collapse or quantum theory are non- local.* The issue needs clarification. We start by imagining an electron hitting a screen after passing a slit. Following the description just deduced, the collapse process proceeds schematically as depicted in Fig- ure 73. An animation that includes another example of a collapse process – inspired by Page 154 Bohm’s thought experiment – can be seen in the lower left corners on these pages, start- ing at page 115. The collapse process has a surprising side: due to the shortness of the decoherence time, during this (and any other) wave function collapse the maximum of the wave function usually changes position faster than light. Is this reasonable? A situation is called acausal or non-local if energy is transported faster than light. Challenge 127 s Using Figure 73 you can determine the energy velocity involved, using the results on Vol. III, page 133 signal propagation. The result is a value smaller than 𝑐. A wave function whose maximum moves faster than light does not automatically imply that energy moves faster than light. In other words, quantum theory contains speeds greater than light, but no energy Ref. 108 speeds greater than light. In classical electrodynamics, the same happens with the scalar and the vector potentials if the Coulomb gauge is used. We have also encountered speeds faster than that of light in the motion of shadows and scissors, and in many other Vol. II, page 58 observations. Any physicist now has two choices: he can be straight, and say that there is no non-locality in nature; or he can be less straight, and claim there is. In the latter * This continues a topic that we know already: we have explored a different type of non-locality, in general Vol. II, page 284 relativity, earlier on. 154 7 superpositions and probabilities space detector 2 detector 1 Motion Mountain – The Adventure of Physics collapse time F I G U R E 74 Bohm’s thought experiment. case, he has to claim that even classical physics is non-local. However, nobody dares to claim this. In fact, there is a danger in this more provoking usage of the term ‘non-local’: copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net a small percentage of those who claim that the world is non-local after a while start to believe that there really is faster-than-light energy transport in nature. These people be- come prisoners of their muddled thinking. On the other hands, muddled thinking helps to get more easily into newspapers. In short, even though the definition of non-locality is not unanimous, here we stick to the stricter one, and define non-locality as energy transport faster than light. An often cited thought experiment that shows the pitfalls of non-locality was pro- posed by Bohm* in the discussion around the so-called Einstein–Podolsky–Rosen Ref. 109, Ref. 110 paradox. In the famous EPR paper the three authors tried to find a contradiction between quantum mechanics and common sense. Bohm translated their rather confused paper into a clear thought experiment that is shown schematically in Figure 74. When two particles in a spin 0 state move apart, measuring one particle’s spin orientation implies an immediate collapse also of the other particle’s spin, namely in the exactly opposite dir- ection. This happens instantaneously over the whole separation distance; no speed limit is obeyed. In other words, entanglement seems to lead to faster-than-light communica- tion. However, in Bohm’s experiment, no energy is transported faster than light. No non- locality is present, despite numerous claims of the contrary by certain authors. The two * David Joseph Bohm (1917–1992), was an influential physicist. He codiscovered the Aharonov–Bohm effect and spent a large part of his later life investigating the connections between quantum physics and philo- sophy. 7 quantum theory without ideology 155 entangled electrons belong to one system: assuming that they are separate only because the wave function has two distant maxima is a conceptual mistake. In fact, no signal can be transmitted with this method; the decoherence is a case of prediction which looks like a signal without being one. Bohm’s experiment, like any other EPR-like experiment, does not allow communication faster than light. We already discussed such cases in the Vol. III, page 136 section on electrodynamics. Bohm’s experiment has actually been performed. The first and most famous realiz- Ref. 111 ation was due, in 1982, by Alain Aspect; he used photons instead of electrons. Like all latter tests, it has fully confirmed quantum mechanics. In fact, experiments such as the one by Aspect confirm that it is impossible to treat either of the two particles as a system by itself; it is impossible to ascribe any physical property, such as a spin orientation, to either of them alone. (The Heisenberg picture would express this restriction even more clearly.) Only the two electrons together form a physical system, because only the pair interacts incoherently with the environment. The mentioned two examples of apparent non-locality can be dismissed with the re- Motion Mountain – The Adventure of Physics mark that since obviously no energy flux faster than light is involved, no problems with causality appear. Therefore the following example is more interesting. Take two identical atoms, one in an excited state, one in the ground state, and call 𝑙 the distance that separ- ates them. Common sense tells that if the first atom returns to its ground state emitting a photon, the second atom can be excited only after a time 𝑡 = 𝑙/𝑐 has been elapsed, i.e., after the photon has travelled to the second atom. Surprisingly, this conclusion is wrong. The atom in its ground state has a non-zero probability to be excited at the same moment in which the first is de-excited. This has Ref. 112 been shown most simply by Gerhard Hegerfeldt. The result has also been confirmed ex- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net perimentally. More careful studies show that the result depends on the type of superposition of the two atoms at the beginning: coherent or incoherent. For incoherent superpositions, the intuitive result is correct; the counter-intuitive result appears only for coherent superpos- itions. Again, a careful discussion shows that no real non-locality of energy is involved. In summary, faster-than-light speeds in wave function collapse do not contradict the limit on energy speed of special relativity. Collapse speeds are phase velocities. In nature, phase velocities are unlimited; unlimited phase velocities never imply energy transport faster than light. In addition, we recover the result that physical systems are only clearly defined if they interact incoherently with their environment. Curiosities and fun challenges ab ou t superpositions Some people wrongly state that atoms in a superposition of two states centred at different positions can be photographed. (This lie is even used by some sects to attract believers.) Challenge 128 s Why is this not true? ∗∗ In a few cases, the superposition of different macroscopic states can actually be observed by lowering the temperature to sufficiently small values and by carefully choosing suit- ably small masses or distances. Two well-known examples of coherent superpositions are those observed in gravitational wave detectors and in Josephson junctions. In the 156 7 superpositions and probabilities Ref. 102 first case, one observes a mass as heavy as 1000 kg in a superposition of states located at different points in space: the distance between them is of the order of 10−17 m. In the second case, in superconducting rings, superpositions of a state in which a macro- scopic current of the order of 1 pA flows in clockwise direction with one where it flows Ref. 113 in counter-clockwise direction have been produced. ∗∗ Ref. 114 Superpositions of magnetization in up and down direction at the same time have been observed for several materials. ∗∗ Since the 1990s, the sport of finding and playing with new systems in coherent mac- Ref. 115 roscopic superpositions has taken off across the world. The challenges lie in the clean experiments necessary. Experiments with single atoms in superpositions of states are Ref. 116 among the most popular ones. Motion Mountain – The Adventure of Physics ∗∗ Ref. 117 In 1997, coherent atom waves were extracted from a cloud of sodium atoms. ∗∗ Macroscopic objects usually are in incoherent states. This is the same situation as for light. The world is full of ‘macroscopic’, i.e., incoherent light: daylight, and all light from lamps, from fire and from glow-worms is incoherent. Only very special and carefully constructed sources, such as lasers or small point sources, emit coherent light. Only these copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net sources allow studying interference effects. In fact, the terms ‘coherent’ and ‘incoherent’ originated in optics, since for light the difference between the two, namely the capacity to interfere, had been observed centuries before the case of matter. Coherence and incoherence of light and of matter manifest themselves differently, be- cause matter can stay at rest but light cannot and because matter is made of fermions, Page 139 but light is made of bosons. Coherence can be observed easily in systems composed of bosons, such as light, sound in solids, or electron pairs in superconductors. Coherence is less easily observed in systems of fermions, such as systems of atoms with their elec- tron clouds. However, in both cases a decoherence time can be defined. In both cases coherence in many particle systems is best observed if all particles are in the same state (superconductivity, laser light) and in both cases the transition from coherent to incoher- ent is due to the interaction with a bath. A beam is thus incoherent if its particles arrive randomly in time and in frequency. In everyday life, the rarity of observation of coherent matter superpositions has the same origin as the rarity of observation of coherent light. ∗∗ We will discuss the relation between the environment and the decay of unstable systems Vol. V, page 47 later on. The phenomenon is completely described by decoherence. ∗∗ Challenge 129 ny Can you find a method to measure the degree of entanglement? Can you do so for a system made of many particles? 7 quantum theory without ideology 157 ∗∗ The study of entanglement leads to a simple conclusion: teleportation contradicts correl- Challenge 130 ny ation. Can you confirm the statement? ∗∗ Challenge 131 s Are ghost images in TV sets, often due to spurious reflections, examples of interference? ∗∗ Challenge 132 d What happens when two monochromatic electrons overlap? ∗∗ Some people say that quantum theory could be used for quantum computing, by using Ref. 118 coherent superpositions of wave functions. Can you give a general reason that makes this aim very difficult – even though not impossible – even without knowing how such Challenge 133 s a quantum computer might work, or what the so-called qubits might be? Motion Mountain – The Adventure of Physics Why d o probabilities and wave function collapse appear in measurements? Measurements in quantum mechanics are puzzling also because they lead to statements in which probabilities appear. For example, we speak about the probability of finding an electron at a certain distance from the nucleus of an atom. Statements like this be- long to the general type ‘when the observable 𝐴 is measured, the probability to find the outcome 𝑎 is 𝑝.’ In the following we will show that the probabilities in such statements are inevitable for any measurement, because, as we will show, (1) any measurement and copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net any observation is a special case of decoherence or disentanglement process and (2) all decoherence processes imply the quantum of action. (Historically, the process of meas- urement was studied before the more general process of decoherence. That explains in part why the topic is so confused in many peoples’ minds.) Vol. III, page 267 What is a measurement? As already mentioned earlier on, a measurement is any in- teraction which produces a record or a memory. (Any effect of everyday life is a record; but this is not true in general. Can you give some examples of effects that are records and Challenge 134 s some effects which are not?) Measurements can be performed by machines; when they are performed by people, they are called observations. In quantum theory, the process of measurement is not as straightforward as in classical physics. This is seen most strikingly when a quantum system, such as a single electron, is first made to pass a diffraction slit, or better – in order to make its wave aspect become apparent – a double slit and then is made to hit a photographic plate, in order to make also its particle aspect appear. Ex- periment shows that the blackened dot, the spot where the electron has hit the screen, cannot be determined in advance. (The same is true for photons or any other particle.) However, for large numbers of electrons, the spatial distribution of the black dots, the so-called diffraction pattern, can be calculated in advance with high precision. The outcome of experiments on microscopic systems thus forces us to use probabil- ities for the description of microsystems. We find that the probability distribution 𝑝(𝑥) of the spots on the photographic plate can be calculated from the wave function 𝜓 of the electron at the screen surface and is given by 𝑝(𝑥) = |𝜓† (𝑥)𝜓(𝑥)|2 . This is in fact a special 158 7 superpositions and probabilities ball gravity pegs F I G U R E 75 A system showing probabilistic behaviour: ball falling through an array of pegs. case of the general first property of quantum measurements: ⊳ The measurement of an observable 𝐴 for a system in a state 𝜓 gives as result Motion Mountain – The Adventure of Physics one of the eigenvalues 𝑎𝑛 , and the probability 𝑃𝑛 to get the result 𝑎𝑛 is given by 𝑃𝑛 = |𝜑𝑛† 𝜓|2 , (79) where 𝜑𝑛 is the eigenfunction of the operator 𝐴 corresponding to the eigen- value 𝑎𝑛 .* Experiments also show a second property of quantum measurements: copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ⊳ After a measurement, the observed quantum system is in the state 𝜑𝑛 cor- responding to the measured eigenvalue 𝑎𝑛 . One also says that during the Ref. 119 measurement, the wave function has collapsed from 𝜓 to 𝜑𝑛 . These two experimental properties can also be generalized to the more general cases with degenerate and continuous eigenvalues. Obviously, the experimental results on the measurement process require an explan- ation. At first sight, the sort of probabilities encountered in quantum theory are differ- ent from the probabilities we encounter in everyday life. Take roulette, dice, the system shown in Figure 75, pachinko machines or the direction in which a pencil on its tip falls: all have been measured experimentally to be random (assuming no cheating by the designer or operators) to a high degree of accuracy. These everyday systems do not puzzle us. We unconsciously assume that the random outcome is due to the small, but uncontrollable variations of the starting conditions or the environment every time the experiment is repeated.** * All linear transformations transform some special vectors, called eigenvectors (from the German word eigen meaning ‘self’) into multiples of themselves. In other words, if 𝑇 is a transformation, 𝑒 a vector, and 𝑇(𝑒) = 𝜆𝑒 (80) where 𝜆 is a scalar, then the vector 𝑒 is called an eigenvector of 𝑇, and 𝜆 is associated eigenvalue. The set of all eigenvalues of a transformation 𝑇 is called the spectrum of 𝑇. ** To get a feeling for the limitations of these unconscious assumptions, you may want to read the already 7 quantum theory without ideology 159 But microscopic systems seem to be different. The two properties of quantum meas- urements just mentioned express what physicists observe in every experiment, even if the initial conditions are taken to be exactly the same every time. But why then is the position for a single electron, or most other observables of quantum systems, not pre- dictable? In other words, what happens during the collapse of the wave function? How long does the collapse take? In the beginning of quantum theory, there was the percep- tion that the observed unpredictability is due to the lack of information about the state of the particle. This lead many to search for so-called ‘hidden variables’. All these at- tempts were doomed to fail, however. It took some time for the scientific community to realize that the unpredictability is not due to the lack of information about the state of the particle, which is indeed described completely by the state vector 𝜓. In order to uncover the origin of probabilities, let us recall the nature of a measure- ment, or better, of a general observation. ⊳ Any observation is the production of a record. Motion Mountain – The Adventure of Physics The record can be a visual or auditive memory in our brain, or a written record on paper, Vol. III, page 265 or a tape recording, or any such type of object. As explained in the previous volume, an object is a record if it cannot have arisen or disappeared by chance. To avoid the influence of chance, all records have to be protected as much as possible from the external world; e.g. one typically puts archives in earthquake safe buildings with fire protection, keeps documents in a safe, avoids brain injury as much as possible, etc. On top of this, records have to be protected from their internal fluctuations. These internal fluctuations are due to the many components any recording device is made of. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net If the fluctuations were too large, they would make it impossible to distinguish between the possible contents of a memory. Now, fluctuations decrease with increasing size of a system, typically with the square root of the size. For example, if a hand writing is too small, it is difficult to read if the paper gets brittle; if the magnetic tracks on tapes are too small, they demagnetize and lose the stored information. In other words, a record is rendered stable against internal fluctuations by making it of sufficient size. Every record thus consists of many components and shows small fluctuations. The importance of size can be expressed in another way: every system with memory, i.e., every system capable of producing a record, contains a bath. In summary, the state- ment that any observation is the production of a record can be expressed more precisely as: ⊳ Any observation of a system is the result of an interaction between that sys- tem and a bath in the recording apparatus. By the way, since baths imply friction, we can also say: memory needs friction. In ad- dition, any observation measuring a physical quantity uses an interaction depending on that same quantity. With these seemingly trivial remarks, we can describe in more detail mentioned story of those physicists who built a machine that could predict the outcome of a roulette ball Vol. I, page 127 from the initial velocity imparted by the croupier. 160 7 superpositions and probabilities the quantum apparatus, e.g. eye, ear, mechanical or machine, with memory, system i.e. coupled to a bath H H int tr describes is determined describes its its possible by the type friction, e.g. states of measurement due to heat flow F I G U R E 76 The concepts used in the description of measurements. Motion Mountain – The Adventure of Physics the process of observation, or, as it is usually called in the quantum theory, the measure- ment process. Any measurement apparatus, or detector, is characterized by two main aspects, shown in Figure 76: the interaction it has with the microscopic system, and the bath it contains Ref. 120 to produce the record. Any description of the measurement process thus is the descrip- tion of the evolution of the microscopic system and the detector; therefore one needs the Hamiltonian for the particle, the interaction Hamiltonian, and the bath properties copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net (such as the relaxation time 𝑡r ). The interaction specifies what is measured and the bath realizes the memory. We know that only classical thermodynamic systems can be irreversible; quantum systems are not. We therefore conclude: a measurement system must be described clas- sically: otherwise it would have no memory and would not be a measurement system: it would not produce a record! Memory is a classical effect. (More precisely, memory is an effect that only appears in the classical limit.) Nevertheless, let us see what happens if we describe the measurement system quantum mechanically. Let us call 𝐴 the observable which is measured in the experiment and its eigen- functions 𝜑𝑛 . We describe the quantum mechanical system under observation – often a particle – by a state 𝜓. The full state of the system can always be written as 𝜓 = 𝜓𝑝 𝜓other = ∑ 𝑐𝑛 𝜑𝑛 𝜓other . (81) 𝑛 Here, 𝜓𝑝 is the aspect of the (particle or system) state that we want to measure, and 𝜓other represents all other degrees of freedom, i.e., those not described – spanned, in mathem- atical language – by the operator 𝐴 corresponding to the observable we want to measure. The numbers 𝑐𝑛 = |𝜑𝑛† 𝜓𝑝 | give the expansion of the state 𝜓𝑝 , which is taken to be nor- malized, in terms of the basis 𝜑𝑛 . For example, in a typical position measurement, the functions 𝜑𝑛 would be the position eigenfunctions and 𝜓other would contain the inform- ation about the momentum, the spin and all other properties of the particle. 7 quantum theory without ideology 161 How does the system–detector interaction look like? Let us call the state of the ap- paratus before the measurement 𝜒start . The measurement apparatus itself, by definition, is a device which, when it is hit by a particle in the state 𝜑𝑛 𝜓other , changes from the state 𝜒start to the state 𝜒𝑛 . One then says that the apparatus has measured the eigenvalue 𝑎𝑛 corresponding to the eigenfunction 𝜑𝑛 of the operator 𝐴. The index 𝑛 is thus the record of the measurement; it is called the pointer index or variable. This index tells us in which state the microscopic system was before the interaction. The important point, taken from our previous discussion, is that the states 𝜒𝑛 , being records, are macroscopically distinct, precisely in the sense of the previous section. Otherwise they would not be records, and the interaction with the detector would not be a measurement. Of course, during measurement, the apparatus sensitive to 𝜑𝑛 changes the part 𝜓other of the particle state to some other situation 𝜓other,𝑛 , which depends on the measurement and on the apparatus; we do not need to specify it in the following discussion.* But let us have an intermediate check of our reasoning. Do apparatuses as described here exist? Yes, they do. For example, any photographic plate is a detector for the position of ion- Motion Mountain – The Adventure of Physics izing particles. A plate, and in general any apparatus measuring position, does this by changing its momentum in a way depending on the measured position: the electron on a photographic plate is stopped. In this case, 𝜒start is a white plate, 𝜑𝑛 would be a particle localized at spot 𝑛, 𝜒𝑛 is the function describing a plate blackened at spot 𝑛 and 𝜓other,n describes the momentum and spin of the particle after it has hit the photographic plate at the spot 𝑛. Now we are ready to look at the measurement process itself. For the moment, let us disregard the bath in the detector, and let us just describe it with a state as well, which we call 𝜒start . In the time before the interaction between the particle and the detector, the copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net combined system (including the detector) was in the initial state 𝜓𝑖 given simply by 𝜓𝑖 = 𝜓𝑝 𝜒start = ∑ 𝑐𝑛 𝜑𝑛 𝜓other 𝜒start , (84) 𝑛 where 𝜓𝑝 is the (particle or system) state. After the interaction, using the just mentioned, experimentally known characteristics of the apparatus, the combined state 𝜓𝑎 is 𝜓𝑎 = ∑ 𝑐𝑛 𝜑𝑛𝜓other,𝑛 𝜒𝑛 . (85) 𝑛 This evolution from 𝜓𝑖 to 𝜓𝑎 follows from the evolution equation applied to the particle– detector combination. Now, the combined state 𝜓𝑎 is a superposition of macroscopically * How does the interaction look like mathematically? From the description we just gave, we specified the final state for every initial state. Since the two density matrices are related by 𝜌f = 𝑇𝜌i 𝑇† (82) Challenge 135 ny we can deduce the Hamiltonian from the matrix 𝑇. Are you able to see how? By the way, one can say in general that an apparatus measuring an observable 𝐴 has a system interaction Hamiltonian depending on the pointer variable 𝐴, and for which one has [𝐻 + 𝐻int , 𝐴] = 0 . (83) 162 7 superpositions and probabilities distinct states: it is a superposition of distinct macroscopic states of the detector. In our example 𝜓𝑎 could correspond to a superposition of one state where a spot on the left upper corner is blackened on an otherwise white plate with another state where a spot on the right lower corner of the otherwise white plate is blackened. Such a situation is never observed. Let us see why. The density matrix 𝜌𝑎 of the combined state 𝜓𝑎 after the measurement given by 𝜌𝑎 = 𝜓𝑎 ⊗ 𝜓𝑎† = ∑ 𝑐𝑛 𝑐𝑚∗ (𝜑𝑛 𝜓other,𝑛 𝜒𝑛 ) ⊗ (𝜑𝑚 𝜓other,𝑚 𝜒𝑚 )† , (86) 𝑛,𝑚 contains large non-diagonal terms, i.e., terms for 𝑛 ≠ 𝑚, whose numerical coefficients are different from zero. Now let us take the bath back in. From the previous section we know the effect of a bath on such a macroscopic superposition. We found that a density matrix such as 𝜌𝑎 decoheres extremely rapidly. We assume here that the decoherence time is negligibly small.* After decoherence, the off-diagonal terms vanish, and only the Motion Mountain – The Adventure of Physics final, diagonal density matrix 𝜌f , given by 𝜌f = ∑|𝑐𝑛 |2 (𝜑𝑛𝜓other,𝑛 𝜒𝑛 ) ⊗ (𝜑𝑛𝜓other,𝑛 𝜒𝑛 )† (87) 𝑛 remains and has experimental relevance. As explained above, such a density matrix de- scribes a mixed state, and the numbers 𝑃𝑛 = |𝑐𝑛 |2 = |𝜑𝑛† 𝜓𝑝 |2 give the probability of meas- uring the value 𝑎𝑛 and of finding the particle in the state 𝜑𝑛 𝜓other,n as well as the detector in the state 𝜒𝑛 . But this is precisely what the two properties of quantum measurements state. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net We therefore find that describing a measurement as an evolution of a quantum sys- tem interacting with a macroscopic detector, itself containing a bath, we can deduce the two properties of quantum measurements, probabilistic outcomes and the collapse of the wave function, from the quantum mechanical evolution equation. The decoherence time 𝑡d of the previous section becomes the time of collapse for the case of a measurement; in addition we find 𝑡collapse = 𝑡d < 𝑡r . (88) In other words, the collapse time is always smaller than the relaxation time of the bath. We thus have a formula for the time the wave function takes to collapse. All experimental Ref. 104 measurements of the time of collapse have confirmed this result. Why is ℏ necessary for probabilities? At first sight, one could argue that the two properties of quantum measurements do not contain ℏ, and thus are not consequences of quantum theory. However, this argument is incorrect. * Note however, that an exactly vanishing decoherence time, which would mean a strictly infinite number of degrees of freedom of the bath or the environment, is in contradiction with the evolution equation, and in particular with unitarity, locality and causality. It is essential in the whole argument not to confuse the logical consequences of a extremely small decoherence time with those of an exactly vanishing decoherence time. 7 quantum theory without ideology 163 Decoherence is a quantum process, because ℏ appears in the expression of the deco- herence time. Since the collapse of the wave function is based on decoherence, it is a quantum process as well. Also probabilities are due to the quantum of action. In addition, we have seen that the concept of wave function appears only because the Page 87 quantum of action ℏ is not zero. Wave functions, their collapse and probabilities are due to the quantum of change ℏ. Page 32 These results recall a statement made earlier on: probabilities appear whenever an experiment attempts to detect changes, i.e., action values, smaller than ℏ. Most puzzles Challenge 136 e around measurement are due to such attempts. However, nature does not allow such measurements; in every such attempt, probabilities appear. Hidden variables A large number of people are not satisfied with the explanation of probabilities in the quantum world. They long for more mystery in quantum theory. They do not like the idea that probabilities are due to baths and to the quantum of action. The most famous Motion Mountain – The Adventure of Physics prejudice such people cultivate is the idea that the probabilities are due to some hidden aspect of nature which is still unknown to humans. Such imagined, unknown aspects are called hidden variables. The beautiful thing about quantum mechanics is that it allows both conceptual and experimental tests on whether such hidden variables exist – without the need of knowing them. Obviously, hidden variables controlling the evolution of microscopic system would contradict the statement that action values below ℏ cannot be detected. The smallest ob- servable action value is the reason for the random behaviour of microscopic systems. The smallest action thus excludes hidden variables. But let us add some more detailed copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net arguments. Historically, the first, somewhat abstract argument against hidden variables was given by John von Neumann.* An additional no-go theorem for hidden variables was pub- Ref. 121 lished by Kochen and Specker in 1967, and independently by John Bell in 1969. The the- orem states: ⊳ Non-contextual hidden variables are impossible, if the Hilbert space has a dimension equal or larger than three. The theorem is about non-contextual variables, i.e., about hidden variables inside the quantum mechanical system. The Kochen–Specker theorem thus states that there is no non-contextual hidden variables model, because mathematics forbids it. This result es- sentially eliminates all possibilities for hidden variables, because usual quantum mech- anical systems have Hilbert space dimensions larger than three. * János Neumann (b. 1903 Budapest, d. 1957 Washington DC) influential mathematician. One of the greatest and clearest scientific minds of the twentieth century, he settled many issues, especially in applied math- ematics and quantum theory, that others still struggle with today. He then worked on the atomic and the hydrogen bomb, on ballistic missiles, and on general defence problems. For the bomb research, he strongly influenced the building of the earliest electronic computers, extending the ideas of Konrad Zuse. At the end of his life, he wanted to change the weather with nuclear bombs. He died of a cancer that was due to his exposure to nuclear radiation when watching bomb tests. 164 7 superpositions and probabilities We cannot avoid noting that there are no restricting theorems about contextual hid- den variables, i.e., variables in the environment and in particular, in the baths contained in it. Indeed, their necessity was shown above! Also common sense eliminates hidden variables, without any recourse to mathemat- ics, with a simple argument. If a quantum mechanical system had internal hidden vari- ables, the measurement apparatus would have zillions of them.* And this would mean that it could not work as a measurement system. Despite all arguments, researchers have always been looking for experimental tests on hidden variables. Most tests are based on the famed Bell’s inequality, a beautifully simple relation published by John Bell** in the 1960s. Can we distinguish quantum theory and locally realistic theories that use hidden vari- ables? Bell’s starting idea is to do so by measuring the polarizations of two correlated photons. Quantum theory says that the polarization of the photons is fixed only at the time it is measured, whereas local realistic models – the most straightforward type of hidden variable models – claim that the polarization is fixed already in advance by a Motion Mountain – The Adventure of Physics hidden variable. As Bell found out, experiments can be used to decide which alternative is correct. Imagine that the polarization is measured at two distant points 𝐴 and 𝐵. Each observer can measure 1 or −1 in each of his favourite direction. Let each observer choose two directions, 1 and 2, and call their results 𝑎1 , 𝑎2 , 𝑏1 and 𝑏2 . Since the measurement results all are either 1 or −1, the value of the specific expression (𝑎1 + 𝑎2 )𝑏1 + (𝑎2 − 𝑎1 )𝑏2 has always the value ±2. Ref. 122 Imagine that you repeat the experiment many times, assuming that the hidden vari- ables appear statistically. You then can deduce (a special case of) Bell’s inequality for two copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Challenge 137 e hidden variables; it predicts that |(𝑎1 𝑏1 ) + (𝑎2 𝑏1 ) + (𝑎2 𝑏2 ) − (𝑎1 𝑏2 )| ⩽ 2 . (89) Here, the expressions in brackets are the averages of the measurement products over a large number of samples. This hidden variable prediction holds independently of the directions of the involved polarizers. On the other hand, for the case that the polarizers 1 and 2 at position 𝐴 and the corresponding ones at position 𝐵 are chosen with angles of π/4, quantum theory predicts that |(𝑎1 𝑏1 ) + (𝑎2 𝑏1 ) + (𝑎2 𝑏2 ) − (𝑎1 𝑏2 )| = 2√2 > 2 . (90) This prediction is in complete contradiction with the hidden variable prediction. Now, all experimental checks of Bell’s inequality have confirmed standard quantum mechanics and falsified hidden variables. There are no exceptions. Another measurable contradiction between quantum theory and locally realistic the- ories has been predicted by Greenberger, Horn and Zeilinger in systems with three en- Ref. 123 tangled particles. Again, quantum theory has been confirmed in all experiments. * Which leads to the definition: one zillion is 1023 . ** John Stewart Bell (1928–1990), theoretical physicist who worked mainly on the foundations of quantum theory. 7 quantum theory without ideology 165 In summary, hidden variables do not exist. Of course, this is not really surprising. The search for hidden variables is based on a misunderstanding of quantum mechanics or on personal desires on how the world should be, instead of taking it as it is: there is a smallest measurable action value, ℏ, in nature. Summary on probabilities and determinism “ Geometrica demonstramus quia facimus; si ” physica demonstrare possemus, faceremus. Giambattista Vico* We draw a number of conclusions which we need for the rest of our mountain ascent. Note that these conclusions, even though in agreement with all experiments, are not yet shared by all physicists! The whole topic is a problem for people who prefer ideology to facts. In everyday life, probabilities often do not appear or are not noted. Quantum theory Motion Mountain – The Adventure of Physics shows: ⊳ Probabilities appear whenever a process tries to distinguish between situ- ations that differ by about one quantum of action ℏ. ⊳ The precise mechanism for the appearance of probabilities is due to the in- volved baths. In short: probabilities appear whenever an experiment tries to distinguish between close situations. In more detail: copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net — Probabilities do not appear in measurements because the state of the quantum sys- tem is unknown or fuzzy, but because the detailed state of an interacting bath in the environment is unknown. Quantum mechanical probabilities are of statistical origin and are due to baths in the environment or in the measurement apparatus, in com- bination with the quantum of action ℏ. The probabilities are due to the large number of degrees of freedom contained in the bath. These large numbers make the outcome of experiments – especially those whose possible outcomes differ by about ℏ – unpre- dictable. If the state of the involved bath were known, the outcome of an experiment could be predicted. The probabilities of quantum theory are due to the quantum of action and are ‘thermodynamic’ in origin. In other words, there are no fundamental probabilities in nature. All probabilities in nature are due to decoherence; in particular, all probabilities are due to the statistics of the many particles – some of which may even be virtual – that are part of the baths in the environment. Modifying well-known words by Albert Einstein, we can agree on the following: ‘nature does not play dice.’ Therefore we called 𝜓 the wave function – instead of ‘probability amplitude’, as is often done. An even better term would be state function. * ‘We are able to demonstrate geometrical matters because we make them; if we could prove physical mat- ters we would be able to make them.’ Giovanni Battista Vico (b. 1668 Napoli, d. 1744 Napoli) important philosopher and thinker. In this famous statement he points out a fundamental distinction between math- ematics and physics. 166 7 superpositions and probabilities — Every observation in everyday life is a special case of decoherence. What is usually called the ‘collapse of the wave function’ is a decoherence process due to the in- teraction with the baths present in the environment or in the measuring apparatus. Because humans are warm-blooded and have memory, humans themselves are meas- urement apparatuses. The fact that our body temperature is 37°C is thus the reason that we see only a single world, and no superpositions. (Actually, there are many ad- Challenge 138 s ditional reasons; can you name a few?) — Every measurement is complete when the microscopic system has interacted with the bath in the measuring apparatus. Quantum theory as a description of nature does not require detectors; the evolution equation describes all examples of motion. However, measurements do require the existence of detectors. A detector, or measurement ap- paratus, is a machine that records observations. Therefore, it has to include a bath, i.e., has to be a classical, macroscopic object. In this context one speaks also of a clas- sical apparatus. This necessity of the measurement apparatus to be classical had been already stressed in the very early stages of quantum theory. Motion Mountain – The Adventure of Physics — All measurements, being decoherence processes that involve interactions with baths, are irreversible processes and increase entropy. — Every measurement, like every example of decoherence, is a special case of quantum mechanical evolution, namely the evolution for the combination of a quantum sys- tem, a macroscopic detector and a bath. Since the evolution equation is relativistically invariant, no causality problems appear in measurements; neither do locality prob- lems or logical problems appear. — Since both the evolution equation and the measurement process do not involve quantities other than space-time, Hamiltonians, baths and wave-functions, no other copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net quantity plays a role in measurement. In particular, no human observer nor any con- Vol. III, page 339 sciousness is involved or necessary. Every measurement is complete when the micro- scopic system has interacted with the bath in the apparatus. The decoherence inherent in every measurement takes place even if nobody is looking. This trivial consequence is in agreement with the observations of everyday life, for example with the fact that the Moon is orbiting the Earth even if nobody looks at it.* Similarly, a tree falling in the middle of a forest makes noise even if nobody listens. Decoherence is independ- ent of human observation, of the human mind and of human existence. — In every measurement the quantum system interacts with the detector. Since there is a minimum value for the magnitude of action, every observation influences the ob- served. Therefore every measurement disturbs the quantum system. Any precise de- scription of observations must also include the description of this disturbance. In the present section the disturbance was modelled by the change of the state of the system from 𝜓other to 𝜓other,n . Without such a change of state, without a disturbance of the quantum system, a measurement is impossible. — Since the complete measurement process is described by quantum mechanics, unitar- ity is and remains the basic property of evolution. There are no non-unitary processes in quantum mechanics. — The description of the collapse of the wave function as a decoherence process is an * The opposite view is sometimes falsely attributed to Niels Bohr. The Moon is obviously in contact with Challenge 139 s many radiation baths. Can you list a few? 7 quantum theory without ideology 167 explanation exactly in the sense in which the term ‘explanation’ was defined earlier Vol. III, page 334 on; it describes the relation between an observation and all the other aspects of reality, in this case the bath in the detector or the environment. The collapse of the wave function has been measured, calculated and explained. The collapse is not a question of ‘interpretation’, i.e., of opinion, as unfortunately often is suggested.* — It is not useful to speculate whether the evolution for a single quantum measurement could be determined if the state of the environment around the system were known. Measurements need baths. But a bath is, to an excellent approximation, irreversible and thus cannot be described by a wave function, which behaves reversibly.** In short: ⊳ Quantum mechanics is deterministic. ⊳ Baths are probabilistic. ⊳ Baths are probabilistic because of the quantum of action. Motion Mountain – The Adventure of Physics In summary, there is no irrationality in quantum theory. Whoever uses quantum theory as argument for superstitions, irrational behaviour, new age beliefs or ideologies is guilty of disinformation. The statement by Gell-Mann at the beginning of this chapter is such Page 143 an example. Another is the following well-known, but incorrect statement by Richard Feynman: Ref. 125 ... nobody understands quantum mechanics. Nobel Prizes obviously do not prevent views distorted by ideology. The correct statement copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net is: ⊳ The quantum of action and decoherence are the key to understanding quantum theory. In fact, ℏ and decoherence allow clarifying many other issues. We explore a few interest- ing ones. What is the difference bet ween space and time? Space and time differ. Objects are localized in space but not in time. Why is this the case? In nature, most bath–system interactions are mediated by a potential. All poten- tials are by definition position dependent. Therefore, every potential, being a function of the position 𝑥, commutes with the position observable (and thus with the interaction Hamiltonian). The decoherence induced by baths – except if special care is taken – thus first of all destroys the non-diagonal elements for every superposition of states centred * This implies that the so-called ‘many worlds’ interpretation is wishful thinking. The conclusion is con- Ref. 124 firmed when studying the details of this religious approach. It is a belief system, not based on facts. ** This very strong type of determinism will be very much challenged in the last part of this text, in which it will be shown that time is not a fundamental concept, and therefore that the debate around determinism looses most of its interest. 168 7 superpositions and probabilities at different locations. In short, objects are localized because they interact with baths via potentials. For the same reason, objects also have only one spatial orientation at a time. If the system–bath interaction is spin-dependent, the bath leads to ‘localization’ in the spin variable. This occurs for all microscopic systems interacting with magnets. As a result, macroscopic superpositions of magnetization are almost never observed. Since electrons, protons and neutrons have a magnetic moment and a spin, this conclusion can even be extended: everyday objects are never seen in superpositions of different rotation states because their interactions with baths are spin-dependent. As a counter-example, most systems are not localized in time, but on the contrary exist for very long times, because practically all system–bath interactions do not commute with time. In fact, this is the way a bath is defined to begin with. In short, objects are permanent because they interact with baths. Are you able to find an interaction which is momentum-dependent instead of Challenge 140 s position-dependent? What is the consequence for macroscopic systems? Motion Mountain – The Adventure of Physics In other words, in contrast to general relativity, quantum theory produces a distinc- tion between space and time. In fact, we can define position as the observable that com- mutes with interaction Hamiltonians. This distinction between space and time is due to the properties of matter and its interactions. We could not have deduced this distinction in general relativity. Are we go od observers? Are humans classical apparatuses? Yes, they are. Even though several prominent physi- cists claim that free will and probabilities are related, a detailed investigation shows that copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Ref. 126 this in not the case. Our senses are classical machines because they obey their definition: human senses record observations by interaction with a bath. Our brain is also a clas- sical apparatus: the neurons are embedded in baths. Quantum probabilities do not play a determining role in the brain. Any observing entity, be it a machine or a human being, needs a bath and a memory to record its observations. This means that observers have to be made of matter; an ob- server cannot be made of radiation. Our description of nature is thus severely biased: we describe it from the standpoint of matter. That is a bit like describing the stars by putting Challenge 141 e the Earth at the centre of the universe: we always put matter at the centre of our descrip- Vol. VI, page 83 tion. Can we eliminate this basic anthropomorphism? We will find out as we continue our adventure. What relates information theory, cryptolo gy and quantum theory? Physics means talking about observations of nature. Like any observation, also measure- ments produce information. It is thus possible to translate much (but not all) of quantum theory into the language of information theory. In particular, the existence of a smallest change value in nature implies that the information about a physical system can never be complete, that information transport has its limits and that information can never be fully trusted. The details of these studies form a fascinating way to look at the micro- scopic world. 7 quantum theory without ideology 169 The analogy between quantum theory and information theory becomes even more Ref. 127 interesting when the statements are translated into the language of cryptology. Crypto- logy is the science of transmitting hidden messages that only the intended receiver can decrypt. In our modern times of constant surveillance, cryptology is an important tool to protect personal freedom.* The quantum of action implies that messages can be sent in an (almost) safe way. Listening to a message is a measurement process. Since there is a smallest action ℏ, we can detect whether somebody has tried to listen to a message that we sent. To avoid a man- in-the-middle attack – somebody who pretends to be the receiver and then sends a copy of the message to the real, intended receiver – we can use entangled systems as signals or messages to transmit the information. If the entanglement is destroyed, somebody has listened to the message. Usually, quantum cryptologists use communication systems based on entangled photons. The major issue of quantum cryptology, a large modern research field, is the key dis- tribution problem. All secure communication is based on a secret key that is used to Motion Mountain – The Adventure of Physics decrypt the message. Even if the communication channel is of the highest security – like entangled photons – one still has to find a way to send the communication partner the secret key necessary for the decryption of the messages. Finding such methods is the main aspect of quantum cryptology. However, close investigation shows that all key ex- change methods are limited in their security. In short, due to the quantum of action, nature provides limits on the possibility of sending encrypted messages. The statement of these limits is (almost) equivalent to the statement that change in nature is limited by the quantum of action. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Is the universe a compu ter? The quantum of action provides a limit to secure information exchange. This connection allows us to brush aside several incorrect statements often found in the media. Stating that ‘the universe is information’ or that ‘the universe is a computer’ is as reasonable Vol. VI, page 109 as saying that the universe is an observation or a chewing-gum dispenser. Any expert of motion should beware of these and similarly fishy statements; people who use them either deceive themselves or try to deceive others. Does the universe have a wave function? And initial conditions? The wave function of the universe is frequently invoked in discussions about quantum theory. Various conclusions are deduced from this idea, for example on the irreversibility of time, on the importance of initial conditions, on changes required to quantum theory and much more. Are these arguments correct? Vol. II, page 223 The first thing to clarify is the meaning of ‘universe’. As explained already, the term can have two meanings: either the collection of all matter and radiation, or this collection plus all of space-time. Let us also recall the meaning of ‘wave function’: it describes the * Cryptology consists of the field of cryptography, the art of coding messages, and the field of cryptoana- lysis, the art of deciphering encrypted messages. For a good introduction to cryptology, see the text by Albrecht Beutelspacher, Jörg Schwenk & Klaus-Dieter Wolfenstätter, Moderne Verfahren der Kryptographie, Vieweg 1995. 170 7 superpositions and probabilities state of a system. The state distinguishes two otherwise identical systems; for example, position and velocity distinguish two otherwise identical ivory balls on a billiard table. Alternatively and equivalently, the state describes changes in time. Does the universe have a state? If we take the wider meaning of universe, it does not. Vol. I, page 27 Talking about the state of the universe is a contradiction: by definition, the concept of state, defined as the non-permanent aspects of an object, is applicable only to parts of the universe. We then can take the narrower sense of ‘universe’ – the sum of all matter and radi- ation only – and ask the question again. To determine the state of all matter and radi- ation, we need a possibility to measure it: we need an environment. But the environment of matter and radiation is space-time only; initial conditions cannot be determined since we need measurements to do this, and thus an apparatus. An apparatus is a material sys- tem with a bath attached to it; however, there is no such system outside the universe. In short, quantum theory does not allow for measurements of the universe. Motion Mountain – The Adventure of Physics ⊳ The universe has no state. Beware of anybody who claims to know something about the wave function of the uni- verse. Just ask him Wheeler’s question: If you know the wave function of the universe, why aren’t you rich? Despite this conclusion, several famous physicists have proposed evolution equa- tions for the wave function of the universe. (The best-known is, ironically, the Wheeler– Ref. 128 DeWitt equation.) It seems a silly point, but not one prediction of these equations has been compared to experiment; the arguments just given even make this impossible in copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net principle. Exploring such equations, so interesting it may seem at first sight, must there- fore be avoided if we want to complete our adventure and avoid getting lost in false be- liefs. There are many additional twists to this story. One is that space-time itself, even without matter, might be a bath. This speculation will be shown to be correct in the last volume of this adventure. The result seems to allow speaking of the wave function of the universe. But then again, it turns out that time is undefined at the scales where space-time is an effective bath; this again implies that the concept of state is not applicable there. A lack of ‘state’ for the universe is a strong statement. It also implies a lack of ini- tial conditions! The arguments are precisely the same. This is a tough result. We are so used to think that the universe has initial conditions that we never question the term. (Even in this text the mistake might appear every now and then.) But there are no initial conditions for the universe. We can retain as summary, valid even in the light of the latest research: The universe is not a system, has no wave function and no initial conditions – independently of what is meant by ‘universe’. Chapter 8 C OL OU R S A N D OT H E R I N T E R AC T ION S BET W E E N L IG H T A N D M AT T E R “ ” Rem tene; verba sequentur.** Cato S tones and all other objects have colours. Why? In other words, what is the Motion Mountain – The Adventure of Physics pecific way in which charged quantum particles that are found inside tones and inside all other objects interact with electromagnetic fields? In this chapter, we first give an overview of the various ways that colours in nature result from the quantum of action, i.e., from the interaction between matter quantons and photons. Then we explore the simplest such system: we show how the quantum of action leads to the various colours produced by hydrogen atoms. After this, we discover that the interaction between matter and radiation leads to other surprising effects, especially when special relativity is taken into account. The causes of colour copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Quantum theory explains all colours in nature. Indeed, all the colours that we observe are due to electrically charged particles. More precisely, colours are due to the interactions of charged particles with photons. All colours are thus quantum effects. So far, we have explored the motion of quantons that are described by mass only. Now we study the motion of particles that are electrically charged. The charged particles at the basis of most colours are electrons and nuclei, including their composites, from ions, atoms and molecules to fluids and solids. Many colour issues are still topic of research. For example, until recently it was unclear why exactly asphalt is black. The exact structure of the chemical compounds, the asphaltenes, that produce the very dark brown colour Ref. 130 was unknown. Only recent research has settled this question. In fact, the development of new colourants and colour effects is an important part of modern industry. An overview of the specific mechanisms that generate colour is given in the following Ref. 129 table. The table includes all colours that appear in everyday life. (Can you find one that Challenge 142 s is missing?) ** ‘Know the subject and the words will follow.’ Marcus Porcius Cato, (234–149 bce) or Cato the elder, Roman politician famous for his speeches and his integrity. 172 8 colours and more TA B L E 7 Causes of colour. Colour type Example D e ta i l s Class I: Colours due to simple excitations 1. Incandescence and free charge radiation Carbon arc lamp, hot steel, Colours are due to continuous lightbulb wire, most stars, spectrum emitted by all hot magma, lava, hot melts matter; colour sequence, given by Wien’s rule, is black, red, orange, yellow, white, blue-white (molten lead and silver © Graela) Wood fire, candle Wood and wax flames are yellow due to incandescence if carbon-rich and oxygen-poor Motion Mountain – The Adventure of Physics White fireworks, flashlamp, Due to metals burning to sparklers oxide at high temperature, such as magnesium, zinc, iron, aluminium or zirconium (sparkler © Sarah Domingos) Nuclear reactors, Due to fast free charges: synchroton light sources, Vavilov–Čerenkov radiation is free electron lasers due to speed of particle larger than the speed of light in matter, Bremsstrahlung is due copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net to the deceleration of charged particles (nuclear reactor core under water, courtesy NASA) 2. Atomic gas excitations Red neon lamp, blue argon Colours are due to transitions lamp, UV mercury lamp, between atomic energy levels yellow sodium street (gas discharges © Pslawinski) lamps, most gas lasers, metal vapour lasers, some fluorescence Aurora, triboluminescence In air, blue and red colours are in scotch tape, due to atomic and molecular crystalloluminescence in energy levels of nitrogen, strontium bromate whereas green, yellow, orange colours are due to oxygen (aurora © Jan Curtis) Lightning, arcs, sparks, Colour lines are due to energy coloured fireworks, most levels of highly excited atoms coloured flames, some (flames of K, Cu, Cs, B, Ca electroluminescence © Philip Evans) 8 colours and more 173 TA B L E 7 Causes of colour (continued). Colour type Example D e ta i l s 3. Vibrations and rotations of molecules Bluish water, blue ice when Colours are due to quantized clear, violet iodine, levels of rotation and red-brown bromine, vibrations in molecules (blue yellow-green chlorine, red iceberg © Marc Shandro) flames from CN or blue-green flames from CH, some gas lasers, blue ozone leading to blue and grey evening sky Class II: Colours due to ligand field effects Motion Mountain – The Adventure of Physics 4. Transition metal compounds Green malachite Colours are due to electronic Cu2 CO3 (OH)2 , blue cobalt states of the ions; phosphors oxide, blue azurite are used in cathodes tubes for Cu3 (CO3 )2 (OH)2 , red to TV/computer displays and on brown hematite Fe2 O3 , fluorescent lamp tubes (green green MnO, white malachite on yellow kasolite, a Mn(OH)2 , brown uranium mineral, picture manganite, chrome green width 5 mm, found in copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Cr2 O3 , green Kolwezi, Zaire/Congo, praesodymium, pink © Stephan Wolfsried, europium and yellow television shadow mask photo samarium compounds, © Planemad) piezochromic and thermochromic Cr2 O3 − Al2 O3 UV and electron phosphors, scintillation, some fluorescence, some lasers 5. Transition metal impurities Ruby, emerald, alexandrite, Electronic states of transition perovskites, corresponding metal ions are excited by light lasers and thus absorb specific wavelengths (ruby on calcite from Mogok, Myanmar, picture width 3 cm, © Rob Lavinsky) 174 8 colours and more TA B L E 7 Causes of colour (continued). Colour type Example D e ta i l s Class III: Colours due to molecular orbitals 6. Organic compounds Red haemoglobin in blood, Colours are due to conjugated blue blood haemocyanin, π-bonds, i.e. to alternating green chlorophyll in plants, single and double bonds in yellow or orange carotenes molecules; floral pigments are in carrots, flowers and almost all anthocyanins, yellow autumn leaves, red betalains or carotenes; used in or purple anthocyanins in colourants for foods and berries, flowers and red cosmetics, in textile dyes, in autumn leaves, blue indigo, electrochromic displays, in red lycopene in tomatoes, inks for colour printers, in Motion Mountain – The Adventure of Physics red meat from photosensitizers (narcissus iron-containing © Thomas Lüthi, blood on myoglobin, brown finger © Ian Humes, berries glucosamine in crust of © Nathan Wall, hair courtesy baked food, brown tannins, dusdin) black eumelanin in human skin, hair and eye, iron-rich variation pheomelanin in redheads, black melanin also in cut copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net apples and bananas as well as in movable sacks in chameleons, brown-black asphalt, some fluorescence, chemiluminescence, phosphorescence, halochromism, electrochromism and thermochromism, dye lasers Glow-worms, some Bioluminescence is due to bacteria and fungi, most excited molecules, generally deep-sea fish, octopi, called luciferines (angler fish, jellyfish, and other length 4.5 cm, © Steve deep-sea animals Haddock) 8 colours and more 175 TA B L E 7 Causes of colour (continued). Colour type Example D e ta i l s 7. Inorganic charge transfer Blue sapphire, blue lapis Light induces change of lazuli, green amazonite, position of an electron from brown-black magnetite one atom to another; for Fe3 O4 and most other iron example, in blue sapphire the minerals (colouring basalt transition is between Ti and black, beer bottles brown, Fe impurities; many paint quartz sand yellow, and pigments use charge transfer many other rocks with colours; fluorescent analytical brown or red tones), black reagents are used in molecular graphite, purple medicine and biology Motion Mountain – The Adventure of Physics permanganate, orange (magnetite found in Laach, potassium dichromate, Germany, picture width yellow molybdates, red 10 mm, © Stephan Wolfsried, hematite Fe2 O3 , some sand desert Evelien fluorescence Willemsen) Class IV: Colours due to energy band effects 8. Metallic bands Gold (green in Colours in reflection and in copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net transmission), pyrite, iron, transmission are due to brass, alloys, silver, copper, transitions of electrons ruby glass between overlapping bands (saxophone © Selmer) 9. Pure semiconductor bands Silicon, GaAs, black galena Colours are due to electron PbS, red cinnabar HgS, transitions between separate cadmium yellow CdS, bands; colour series is black, black CdSe, red CdSx Se1−x , red, orange, yellow, white ZnO, orange white/colourless; some used vermillion HgS, colourless as pigments (zinc oxide diamond, black to gold courtesy Walkerma) piezochromic SmS 10. Doped semiconductor bands Blue, yellow, green and Colours are due to transitions black diamond; LEDs; between dopants and semiconductor lasers; solar semiconductor bands cells; ZnS and Znx Cd1−x S (e.g. blue diamond: boron based and other phosphors accepters, black diamond: nitrogen donors) (quantum dots © Andrey Rogach) 176 8 colours and more TA B L E 7 Causes of colour (continued). Colour type Example D e ta i l s 11. Colour centres Amethyst, smoky quartz, Colours are due to colour fluorite, green diamonds, centres, i.e. to electrons or to blue, yellow and brown holes bound at crystal topaz, brown salt, purple vacancies; colour centres are colour of irradiated glass usually created by radiation containing Mn2+ , (amethyst © Rob Lavinsky) lyoluminescence, some fluorescence, F-centre lasers Some light-dependent The photochromic colouring Motion Mountain – The Adventure of Physics sunglasses is due to colour centres formed by the UV light of the Sun Class V: Colours due to physical and geometrical optics 12. Dispersive refraction and polarization Cut diamond, cut zirconia, Spectral decomposition halos and sun dogs formed (sparkle or ‘fire’ of by ice crystals in the air gemstones) is due to dispersion in crystals copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net (zirconia photo © Gregory Phillips) Rainbow Colours of primary and secondary bow are due to dispersion in water droplets Green flash dispersion in the atmosphere shifts the sun colours 13. Scattering Blue sky, blue colouring of Blue light is scattered more distant mountains, red than red light by Rayleigh sunset; colour scattering, when scatterers intensification by (molecules, dust) are smaller pollution; blue quartz than the wavelength of light (Tokyo sunset © Altus Plunkett, blue quartz © David Lynch) 8 colours and more 177 TA B L E 7 Causes of colour (continued). Colour type Example D e ta i l s White colour of hair, milk, The white colour is due to beer foam, clouds, fog, wavelength-independent Mie cigarette smoke coming out scattering, i.e. scattering at of lungs, snow, whipped particles larger than the cream, shampoo, stars in wavelength of light (snow gemstones man © Andreas Kostner) Blue human skin colour in Tyndall blue colours are due cold weather, blue and to scattering on small particles green eyes in humans, blue in front of a dark background Motion Mountain – The Adventure of Physics monkey skin, blue turkey (blue poison frog Dendrobates necks, most blue fish, blue azureus © Lee Hancock) reptiles, blue cigarette smoke Ruby glass The red colour of Murano glass is due to scattering by tiny colloidal gold particles included in the glass in combination with the metallic band structure of gold (ruby copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net glass © murano-glass-shop.it) Nonlinearities, Raman Frequency-shifting scattering, effect, potassium second harmonic generation dihydrogen phosphate and other nonlinearities of (KDP) certain materials change the colour of light impinging with high intensities (800 nm to 400 nm frequency doubling ring laser © Jeff Sherman) 14. Interference (without diffraction) Nacre, oil films, soap Thin film interference bubbles, coatings on produces a standard colour camera lenses, eyes of cats sequence that allows precise in the dark, wings of flies thickness determination and dragonflies, fish scales, (abalone shell © Anne Elliot) some snakes, pearls, tempering colours of steel Polarization colours of thin Colours are due to layers of birefringent interference, as shown by the crystals or thicker layers of dependence on layer stressed polymers thickness (photoelasticity courtesy Nevit Dilmen) 178 8 colours and more TA B L E 7 Causes of colour (continued). Colour type Example D e ta i l s Supernumerary rainbows Due to interference, as shown (see page 102 in volume III) by the dependence on drop size Iridescent beetles, Due to scattering at small butterflies and bird structures or at nanoparticles, feathers, iridescent colours as shown by the angular on banknotes and on cars dependence of the colour (mallard duck © Simon Griffith) 15. Diffraction (with interference) Opal Colours are due to the tiny Motion Mountain – The Adventure of Physics spheres included in the water inside the opal; colours can change if the opal dries out (polished Brazilian opal © Opalsnopals) Aureole, glory, corona Colours are due to diffraction at the tiny mist droplets (aeroplane condensation cloud iridescence © Franz Kerschbaum) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Diffraction gratings, CDs, Colours are due to diffraction vinyl records, some beetles and interference at tiny, and snakes regular pits (CD illuminated by flashlamp © Alfons Reichert) Photonic crystals A modern research topic Cholesteric liquid crystals, Colours are due to diffraction certain beetles and interference in internal material layers (liquid crystal colours © Ingo Dierking) Class VI: Colours due to eye limitations Fechner colours, as on lite.bu. Benham’s wheel or top Colours are due to different edu/vision/applets/Color/ speed response of different Benham/Benham.html photoreceptors 8 colours and more 179 TA B L E 7 Causes of colour (continued). Colour type Example D e ta i l s Internal colour production when Phosphenes Occur through pressure eyes are stimulated (rubbing, sneeze), or with electric or magnetic fields Polarization colours Haidinger’s brush See page 113 in volume III Colour illusions, as on www.psy. Appearing and Effects are due to ritsumei.ac.jp/~akitaoka/color9e. disappearing colours combinations of brain html processing and eye limitations False colour output of eye, as Red light can be seen as Observable with adaptive described on page 196 in volume green optics, if red light is focused III on a green-sensitive cone Colour-blind or ‘daltonic’ Protan, deutan or tritan Each type limits colour person, see page 209 in volume perception in a different way Motion Mountain – The Adventure of Physics III, with reduced colour spectrum Colours fascinate. Fascination always also means business; indeed, a large part of the chemical industry is dedicated to synthesizing colourants for paints, inks, clothes, food and cosmetics. Also evolution uses the fascination of colours for its own business, namely propagating life. The specialists in this domain are the flowering plants. The chemistry of colour production in plants is extremely involved and at least as interesting as the production of colours in factories. Practically all flower colourants, from white, yellow, orange, red to blue, are from three chemical classes: the carotenoids, the anthocyanins copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net (flavonoids) and the betalains. These colourants are stored in petals inside dedicated con- Ref. 131 tainers, the vacuoles. There are many good review articles providing the details. Even though colours are common in plants and animals, most higher animals do not produce many colourants themselves. For example, humans produce only one colour- ant: melanin. (Hemoglobin, which colours blood red, is not a dedicated colourant, but transports the oxygen from the lungs through the body. Also the pink myoglobin in the muscles is not a dedicated colourant.) Many higher animals, such as birds, need to eat the colourants that are so characteristic for their appearance. The yellow colour of legs of pigeons is an example. It has been shown that the connection between colour and nutrition is regularly used by potential mates to judge from the body colours whether a Ref. 132 proposing partner is sufficiently healthy, and thus sufficiently attractive. Above all, the previous table distinguished six main classes among the causes of col- ours. The study of the first class, the colours of incandescence, led Max Planck to discover the quantum of action. In the meantime, research has confirmed that in each class, all colours are due to the quantum of action ℏ. The relation between the quantum of action and the material properties of atoms, molecules, liquids and solids are so well known that colourants can now be designed on the computer. In summary, an exploration of the causes of colours found in nature confirms that all colours are due to quantum effects. We show this by exploring the simplest example: the colours of atomic gas excitations. 180 8 colours and more Motion Mountain – The Adventure of Physics F I G U R E 77 The spectrum of daylight: a stacked image of an extended rainbow, showing its Fraunhofer lines (© Nigel Sharp, NOAO, FTS, NSO, KPNO, AURA, NSF). copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Using the rainb ow to determine what stars are made of Near the beginning of the eighteenth century, Bavarian instrument-maker Joseph Fraunhofer* and the English physicist William Wollaston noted that the rainbow lacks certain colours. These colours appear as black lines when the rainbow is spread out in sufficient breadth. Figure 77 shows the lines in detail; they are called Fraunhofer lines today. In 1860, Gustav Kirchhoff and Robert Bunsen showed that the colours missing in the rainbow were exactly those colours that certain elements emit when heated. In this way they managed to show that sodium, calcium, barium, nickel, magnesium, zinc, cop- per and iron are present in the Sun. Looking at the rainbow thus tells us what the Sun is made of. * Joseph Fraunhofer (b. 1787 Straubing, d. 1826 Munich), having been orphaned at the age of 11, learned lens-polishing. He taught himself optics from books. He entered an optical company at the age of 19, en- suring the success of the business by producing the best available lenses, telescopes, micrometers, optical gratings and optical systems of his time. He invented the spectroscope and the heliometer. He discovered Vol. II, page 312 and counted 476 lines in the spectrum of the Sun; these lines are now named after him. (Today, Fraunhofer lines are still used as measurement standards: the second and the metre are defined in terms of them.) Phys- icists from all over the world would buy their equipment from him, visit him, and ask for copies of his pub- lications. Even after his death, his instruments remained unsurpassed for generations. With his telescopes, in 1837 Bessel was able to make the first measurement of parallax of a star, and in 1846 Johann Gottfried Galle discovered Neptune. Fraunhofer became a professor in 1819. He died young, from the consequences of the years spent working with lead and glass powder. 8 colours and more 181 F I G U R E 78 A low-pressure hydrogen discharge in a 20 cm long glass tube (© Jürgen Bauer at www. smart-elements.com). Of the 476 Fraunhofer lines that Kirchhoff and Bunsen observed, 13 did not corres- pond to any known element. In 1868, Jules Janssen and Joseph Lockyer independently predicted that these unknown lines were from an unknown element. The element was eventually found on Earth, in an uranium mineral called cleveite, in 1895. The new ele- Motion Mountain – The Adventure of Physics ment was called helium, from the Greek word ἥλιος ‘helios’ – Sun. In 1925, using an equation developed by Saha and Langmuir, the young physicist Cecilia Payne (b. 1900 Wendover, England, d. 1979 Cambridge, Massachusetts) taught the world how to deduce the mass percentage of each element from the light spectrum of a star. She did so in her brilliant PhD thesis. Above all, she found that hydrogen and helium were the two most abundant elements in the Sun, in stars, and thus in the whole universe. This went completely against the ideas of the time, but is now common know- ledge. Payne had completed the study of physics in Cambridge, UK, but had not received a degree there because she was a woman. So she left for the United States, where the situ- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ation was somewhat better, and where she worked on her PhD thesis; eventually, she became professor at Harvard University, and later head of its astronomy department. Above all, Payne became an important role model for many female scientists. Despite being the second most common element in the universe, helium is rare on Earth because it is a light noble gas that does not form chemical compounds. Helium atoms on Earth thus rise in the atmosphere and finally escape into space. Understanding the colour lines produced by each element had started to become in- teresting already before the discovery of helium; but afterwards the interest increased further, thanks to the increasing number of applications of colour knowledge in chem- istry, physics, technology, crystallography, biology and lasers. Colours are big business, as the fashion industry, the media and the advertising business show. Vol. III, page 125 In summary, colours are specific mixtures of light frequencies. Light is an electromag- netic wave and is emitted by moving charges. For a physicist, colours thus result from the interaction of charged matter with the electromagnetic field. Now, sharp colour lines cannot be explained by classical electrodynamics. We need quantum theory to explain them. What determines the colours of atoms? The simplest colours to study are the sharp colour lines emitted or absorbed by single atoms. Single atoms are found in gases. The simplest atom to study is that of hydrogen. As shown in Figure 78, hot hydrogen gas emits light. The light consists of a handful of 182 8 colours and more sharp spectral lines that are shown on the left of Figure 79. Already in 1885, the Swiss schoolteacher Johann Balmer (1828–1898) had discovered that the wavelengths of visible hydrogen lines obey the formula: 1 1 1 = 𝑅( − 2) for 𝑚 = 3, 4, 5, ... . (91) 𝜆𝑚 4 𝑚 Careful measurements, which included the hydrogen’s spectral lines in the infrared and in the ultraviolet, allowed Johannes Rydberg (1854–1919) to generalize this formula to: 1 1 1 = 𝑅( 2 − 2) , (92) 𝜆 𝑚𝑛 𝑛 𝑚 where 𝑛 and 𝑚 > 𝑛 are positive integers, and the so-called Rydberg constant 𝑅 has the value 10.97 μm−1 ; easier to remember, the inverse value is 1/𝑅 = 91.16 nm. All the colour Motion Mountain – The Adventure of Physics lines emitted by hydrogen satisfy this simple formula. Classical physics cannot explain this result at all. Thus, quantum theory has a clearly defined challenge here: to explain the formula and the value of 𝑅. Incidentally, the transition 𝜆 21 for hydrogen is called the Lyman-alpha line. Its wavelength, 121.6 nm, lies in the ultraviolet. It is easily observed with telescopes, since most of the visible stars consist of excited hydrogen. The Lyman-alpha line is routinely used to determine the speed of distant stars or galaxies, since the Doppler effect changes Ref. 133 the wavelength when the speed is large. The record in 2004 was a galaxy with a Lyman- alpha line shifted to 1337 nm. Can you calculate the speed with which it is moving away Challenge 143 s from the Earth? copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net From the start, it was clear that the colours of hydrogen are due to the motion of its Challenge 144 e electron. (Why?) The first way to deduce Balmer’s formula from the quantum of action was found by Niels Bohr in 1903. Bohr understood that in contrast to planets circling the Sun, the electron moving around the proton has only a discrete number of possible Page 81 motion states: the angular momentum of the electron is quantized. Assuming that the angular momentum of the electron is an integer multiple of ℏ directly yields Balmer’s Challenge 145 e formula and explains the numerical value of the Rydberg constant 𝑅. This calculation is so famous that it is found in many secondary school books. The result also strengthened Bohr’s decision to dedicate his life to the exploration of the structure of the atom. Twenty years time later, in 1926, Erwin Schrödinger solved his equation of motion for an electron moving in the electrostatic potential 𝑉(𝑟) = 𝑒2 /4π𝜀0 𝑟 of a point-like proton. By doing so, Schrödinger reproduced Bohr’s result, deduced Balmer’s formula and be- came famous in the world of physics. However, this important calculation is long and complex. It can be simplified. In order to understand hydrogen colours, it is not necessary to solve an equation of motion for the electron; it is sufficient to compare the energies of the initial and final states of the electron. This can be done most easily by noting that a specific form of the action must be a multiple of ℏ/2. This approach, a generalization of Bohr’s explanation, was developed by Einstein, Brillouin and Keller, and is now named EBK quantization. It 8 colours and more 183 Hydrogen: spectral lines and energy levels Continuum of ionized states Energy n= 8 954.597 nm 3D5/2 3D5/2 nm n=3 3P3/2 , 3D3/2 3P3/2 , 3D3/2 3S1/2 , 3P1/2 3S1/2 F=2 700 3P1/2 F=1 n=2 2P3/2 2P3/2 F=1 656.2852 nm 2S1/2 , 2P1/2 2S1/2 F=0 656.272 nm 2P1/2 650 F=1 F=0 600 550 Motion Mountain – The Adventure of Physics 500 486.133 nm n=1 450 1S1/2 F=1 434.047 nm 1S1/2 F=0 410.174 nm 400 nonrelativistic relativistic virtual particle nuclear levels 397.007 nm (Bohr) levels (Sommerfeld- levels (with at higher scale copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Dirac) levels Lamb shift) (hyperfine (fine structure) structure) F I G U R E 79 Atomic hydrogen: the visible spectrum of hydrogen (NASA) and its calculated energy levels, in four approximations of increasing precision. Can you associate the visible lines to the correct level transitions? Ref. 134 relies on the fact that the action 𝑆 of any quantum system obeys 1 𝜇 𝑆= ∮ d𝑞𝑖 𝑝𝑖 = (𝑛𝑖 + 𝑖 ) ℏ (93) 2π 4 for every coordinate 𝑞𝑖 and its conjugate momentum 𝑝𝑖 . The expression reflects the simil- arity between angular momentum and action. Here, 𝑛𝑖 can be zero or any positive integer, and 𝜇𝑖 is the so-called Maslov index, an even integer, which in the case of atoms has the value 2 for the radial and azimuthal coordinates 𝑟 and 𝜃, and 0 for the rotation angle 𝜑. The integral is to be taken along a full orbit. In simple words, the action 𝑆 is a half-integer multiple of the quantum of action. This result can be used to calculate the energy levels of periodic quantum systems. Let us do so for hydrogen atoms. Any rotational motion in a spherical potential 𝑉(𝑟) is characterized by a constant energy 𝐸 and constant angular momenta 𝐿 and 𝐿 𝑧. Therefore the conjugate momenta 184 8 colours and more Challenge 146 ny for the coordinates 𝑟, 𝜃 and 𝜑 are 𝐿2 𝑝𝑟 = √2𝑚(𝐸 − 𝑉(𝑟)) − 𝑟2 𝐿2𝑧 𝑝𝜃 = √𝐿2 − sin2 𝜃 𝑝𝜑 = 𝐿 𝑧 . (94) Using these expressions in equation (93) and setting 𝑛 = 𝑛𝑟 + 𝑛𝜃 + 𝑛𝜑 + 1, we get* the result 1 𝑚𝑒4 𝑅ℎ𝑐 𝑐2 𝑚𝛼2 2.19 aJ 13.6 eV 𝐸𝑛 = − 2 2 2 = − 2 = − 2 ≈− 2 ≈− . (97) 𝑛 2(4π𝜀0 ) ℏ 𝑛 2𝑛 𝑛 𝑛2 These energy levels 𝐸𝑛, the non-relativistic Bohr levels, are shown in Figure 79. Using the Motion Mountain – The Adventure of Physics idea that a hydrogen atom emits a single photon when its electron changes from state 𝐸𝑛 to 𝐸𝑚 , we get exactly the formula deduced by Balmer and Rydberg from observations! Challenge 148 e The match between observation and calculation is about four digits. For (almost) the first time ever, a material property, the colour of hydrogen atoms, had been explained from a fundamental principle of nature. Key to this explanation was the quantum of action ℏ. (This whole discussion assumes that the electrons in hydrogen atoms that emit light are Challenge 149 s in eigenstates. Can you argue why this is the case?) In short, the quantum of action implies that only certain specific energy values for an electron are allowed inside an atom. The lowest energy level, for 𝑛 = 1, is called the copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ground state. Its energy value 2.19 aJ is the ionization energy of hydrogen; if that energy is added to the ground state, the electron is no longer bound to the nucleus. The ionization energy thus plays the same role for electrons around atoms as does the escape velocity, or better, the escape energy, for satellites or rockets shot from planets. In the same way that the quantum of action determines the colours of the hydrogen Page 180 atom, it determines the colours of all other atoms. All Fraunhofer lines, whether observed in the infrared, visible or ultraviolet, are due to the quantum of action. In fact, every colour in nature is due to a mixture of colour lines, so that all colours, also those of solids and liquids, are determined by the quantum of action. 8 colours and more 185 Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net F I G U R E 80 The ﬁgure shows the calculated and the measured nodal structure of the hydrogen atom in a weak external electric ﬁeld, magniﬁed by an electrostatic lens. The patterns are two-dimensional interference shadows of the wave functions. Left column: how the wave function is projected from the atoms to the macroscopic screen; central column: the measured nodal structure; right column: comparison of the measured (solid) and calculated (dashed) electron densities. (© Aneta Stodolna/APS, from Ref. 138). The shape of atoms Free atoms are spherical. Atoms in external fields are deformed. Whatever the situation, the shape of atoms is due to the shape of the wave function. The simplest case is the * The calculation is straightforward. After insertion of 𝑉(𝑟) = 𝑒/4π𝜀0 𝑟 into equation (94) one needs to Challenge 147 ny perform the (tricky) integration. Using the general result 1 d𝑧 𝐵 ∮ √𝐴𝑧2 + 2𝐵𝑧 − 𝐶 = −√𝐶 + (95) 2π 𝑧 √−𝐴 one gets 1 𝑒2 √ 𝑚 (𝑛𝑟 + ) ℏ + 𝐿 = 𝑛ℏ = . (96) 2 4π𝜀0 −2𝐸 This leads to the energy formula (97). 186 8 colours and more hydrogen atom. Its wave functions – more precisely, the eigenfunctions for the first few energy levels – are illustrated on the right hand side of Figure 81. These functions had been calculated by Erwin Schrödinger already in 1926 and are found in all textbooks. We do not perform the calculation here, and just show the results. The square of the wave function is the probability density of the electron. This density quickly decreases with increasing distance from the nucleus. Like for a real cloud, the density is never zero, even at large distances. We could thus argue that all atoms have infinite size; in practice however, chemical bonds or the arrangement of atoms in solids show that it is much more appropriate to imagine atoms as clouds of finite size. Surprisingly, the first measurement of the wave function of an atom dates only from the year 2013; it was performed with a clever photoionization technique by Aneta Sto- Ref. 138 dolna and her team. The beautiful experimental result is shown in Figure 80. The figures confirm that wave functions, in contrast to probability densities, have nodes, i.e. lines – or better, surfaces – where their value is zero. In summary, all experiments confirm that the electron in the hydrogen atom forms Motion Mountain – The Adventure of Physics wave functions in exactly the way that is predicted by quantum theory. In particular, the shape of atoms is found to agree with the calculation from quantum mechanics. The size of atoms The calculation of the hydrogen energy levels also yields the effective radius of the elec- tron orbits. It is given by ℏ2 4π𝜀0 ℏ 𝑟𝑛 = 𝑛2 2 = = 𝑛2 𝑎0 ≈ 𝑛2 52.918 937 pm , with 𝑛 = 1, 2, 3, ... (98) 𝑚e 𝑒 𝑚𝑒 𝑐𝛼 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net We again see that, in contrast to classical physics, quantum theory allows only certain specific orbits around the nucleus. (For more details about the fine-structure constant 𝛼, Page 188, page 196 see below.) To be more precise, these radii are the average sizes of the electron clouds surrounding the nucleus. The smallest orbital radius value, 53 pm for 𝑛 = 1, is called the Bohr radius, and is denoted by 𝑎0 . In a gas of hydrogen atoms, most atoms are in the ground state described by 𝑟1 = 𝑎0 and 𝐸1 . On the other hand, quantum theory implies that a hydrogen atom excited to the Ref. 135 level 𝑛 = 500 is about 12 μm in size: larger than many bacteria! Such blown-up atoms, usually called Rydberg atoms, have indeed been observed in the laboratory, although they are extremely sensitive to perturbations. In short, the quantum of action determines the size of atoms. The result thus confirms Page 21 the prediction by Arthur Erich Haas from 1910. In other words ⊳ The quantum of action determines the size of all things. In 1915, Arnold Sommerfeld understood that the analogy of electron motion with orbital gravitational motion could be continued in two ways. First of all, electrons can move, on average, on ellipses instead of circles. The quantization of angular momentum then implies that only selected eccentricities are possible. The higher the angular mo- mentum, the larger the number of possibilities: the first few are shown in Figure 81. The 8 colours and more 187 0 0.2 0.4 0.6 0.8 1 nm nucleus n=1, l=0 (not to scale) n=2, l=1 n=2, l=0 n=3, l=2 n=3, l=1 n=3, l=0 Motion Mountain – The Adventure of Physics F I G U R E 81 The imagined, but not existing and thus false electron orbits of the Bohr–Sommerfeld model of the hydrogen atom (left) and the correct description, using the probability density of the electron in the various states (right) (© Wikimedia). highest eccentricity corresponds to the minimum value 𝑙 = 0 of the so-called azimuthal quantum number, whereas the case 𝑙 = 𝑛 − 1 correspond to circular orbits. Furthermore, copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Ref. 137 the ellipses can have different orientations in space. The second point Sommerfeld noted was that the speeds of the electron in hydrogen are – somewhat – relativistic: the speed values are not negligible compared to the speed of light. Indeed, the orbital frequency of electrons in hydrogen is 1 𝑒4 𝑚e 1 𝑚𝑒 𝑐2 𝛼2 6.7 PHz 𝑓𝑛 = 3 2 3 = 3 ≈ (99) 𝑛 4𝜀0 ℎ 𝑛 ℎ 𝑛3 and the electron speed is 1 𝑒2 𝛼𝑐 2.2 Mm/s 0.007 𝑐 𝑣𝑛 = = ≈ ≈ . (100) 𝑛 4π𝜀0 ℏ 𝑛 𝑛 𝑛 As expected, the further the electron’s orbit is from the nucleus, the more slowly it moves. This result can also be checked by experiment: exchanging the electron for a muon allows us to measure the time dilation of its lifetime. Measurements are in excellent agreement Ref. 136 with the calculations. In short, Sommerfeld noted that Bohr’s calculation did not take into account relativ- istic effects. And indeed, high-precision measurements show slight differences between the Bohr’s non-relativistical energy levels and the measured ones. The calculation must be improved. 188 8 colours and more R elativistic hydro gen Measuring atomic energy levels is possible with a much higher precision than measuring wave functions. In particular, energy level measurements allow to observe relativistic effects. Also in the relativistic case, the EBK action has to be a multiple of ℏ/2. From the re- Ref. 134 lativistic expression for the kinetic energy of the electron in a hydrogen atom 𝑒2 𝐸 + 𝑐2 𝑚 = √𝑝2 𝑐2 + 𝑚2 𝑐4 − (101) 4π𝜀0 𝑟 Challenge 150 e we get the expression 𝐸 2𝑚𝑒2 𝐸 𝑝𝑟2 = 2𝑚𝐸 (1 + 2 ) + (1 + 2 ) . (102) 2𝑐 𝑚 4π𝜀0 𝑟 𝑐𝑚 Motion Mountain – The Adventure of Physics We now introduce, for convenience, the so-called fine-structure constant, as 𝛼 = 𝑒2 /(4π𝜀0 ℏ𝑐) = √4πℏ𝑅/𝑚𝑐 ≈ 1/137.036. (𝛼 is a dimensionless constant; 𝑅 = 10.97 μm−1 Challenge 151 e is the Rydberg constant.) The radial EBK action then implies that 𝑐2 𝑚 𝐸𝑛𝑙 + 𝑐2 𝑚 = . (103) 𝛼2 √1 + 2 (𝑛−𝑙− 12 +√(𝑙+ 12 )2 −𝛼2 ) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net This result, first found by Arnold Sommerfeld in 1915, is correct for point-like, i.e., non- rotating electrons. In reality, the electron has spin 1/2; the correct relativistic energy levels thus appear when we set 𝑙 = 𝑗 ± 1/2 in the above formula. The result can be ap- proximated by 𝑅 𝛼2 𝑛 3 𝐸𝑛𝑗 = − 2 (1 + 2 ( 1 − ) + ...) . (104) 𝑛 𝑛 𝑗+ 2 4 It reproduces the hydrogen spectrum to an extremely high accuracy. If we compare the result with the non-relativistic one, we note that each non-relativistic level 𝑛 is split in Page 183 𝑛 different levels. This splitting is illustrated in Figure 79. In precision experiments, the splitting of the lines of the hydrogen spectrum is visible as the so-called fine structure. The magnitude of the fine structure depends on 𝛼, a fundamental constant of nature. Since Arnold Sommerfeld discovered the importance of this fundamental constant in this con- text, the name he chose, the fine-structure constant, has been taken over across the world. Page 196 The fine-structure constant describes the strength of the electromagnetic interaction; the fine-structure constant is the electromagnetic coupling constant. Modern high-precision experiments show additional effects that modify the colours Page 183 of atomic hydrogen. They are also illustrated in Figure 79. Virtual-particle effects and the coupling of the proton spin give additional corrections. But that is still not all: isotope effects, Doppler shifts and level shifts due to environmental electric or magnetic fields 8 colours and more 189 F I G U R E 82 Paul Dirac (1902–1984) also influence the hydrogen spectrum. The final effect on the hydrogen spectrum, the Vol. V, page 125 famous Lamb shift, will be a topic later on. R elativistic wave equations – again Motion Mountain – The Adventure of Physics “ ” The equation was more intelligent than I was. Paul Dirac about his equation, repeating a statement made by Heinrich Hertz. What is the evolution equation for the wave function in the case that relativity, spin and interactions with the electromagnetic field are taken into account? We could try to gen- Page 106 eralize the representation of relativistic motion given by Foldy and Wouthuysen to the case of particles with electromagnetic interactions. Unfortunately, this is not a simple matter. The simple identity between the classical and quantum-mechanical descriptions is lost if electromagnetism is included. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Charged quantum particles are best described by another, equivalent representation of the Hamiltonian, which was discovered much earlier, in 1926, by the British physicist Paul Dirac.* Dirac found a neat trick to take the square root appearing in the relativistic energy operator. In Dirac’s representation, the Hamilton operator is given by 𝐻Dirac = 𝛽𝑚 + 𝛼 ⋅ 𝑝 . (105) The quantities 𝛽 and the three components (𝛼1 , 𝛼2 , 𝛼3 ) = 𝛼 turn out to be complex 4 × 4 matrices. In Dirac’s representation, the position operator 𝑥 is not the position of a particle, but has additional terms; its velocity operator has only the eigenvalues plus or minus the velocity of light; the velocity operator is not simply related to the momentum operator; * Paul Adrien Maurice Dirac (b. 1902 Bristol, d. 1984 Tallahassee), bilingual physicist, studied electrotech- nics in Bristol, then went to Cambridge, where he later became a professor, holding the chair that Newton had once held. In the years from 1925 to 1933 he published a stream of papers, of which several were worth a Nobel Prize; he received it in 1933. Dirac unified special relativity and quantum theory, predicted antimat- ter, worked on spin and statistics, predicted magnetic monopoles, speculated on the law of large numbers, and more besides. His introversion, friendliness and shyness, and his deep insights into nature, combined with a dedication to beauty in theoretical physics, made him a legend all over the world during his lifetime. For the latter half of his life he tried, unsuccessfully, to find an alternative to quantum electrodynamics, of which he was the founder, as he was repelled by the problems of infinities. He died in Florida, where he lived and worked after his retirement from Cambridge. 190 8 colours and more F I G U R E 83 The famous Zitterbewegung: the superposition of positive and negative energy states leads to an oscillation around a mean vale. Colour indicates phase; Motion Mountain – The Adventure of Physics two coloured curves are shown, as the Dirac equation in one dimension has only two components (not four); the grey curve is the probability density. (QuickTime ﬁlm © Bernd Thaller) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net the equation of motion contains the famous ‘Zitterbewegung’ term; orbital angular mo- mentum and spin are not separate constants of motion. So why use this horrible Hamiltonian? Because only the Dirac Hamiltonian can easily be used for charged particles. Indeed, it is transformed to the Hamiltonian coupled to the Vol. III, page 85 electromagnetic field by the so-called minimal coupling, i.e., by the substitution 𝑝 → 𝑝 − 𝑞𝐴 , (106) that treats electromagnetic momentum like particle momentum. With this prescription, Dirac’s Hamiltonian describes the motion of charged particles interacting with an elec- tromagnetic field 𝐴. The minimal coupling substitution is not possible in the Foldy– Wouthuysen Hamiltonian. In the Dirac representation, particles are pure, point-like, structureless electric charges; in the Foldy–Wouthuysen representation they acquire a Ref. 139 charge radius and a magnetic-moment interaction. (We will come to the reasons below, in the section on QED.) In more detail, the simplest description of an electron (or any other elementary, stable, 8 colours and more 191 electrically-charged particle of spin 1/2) is given by the action 𝑆 and Lagrangian 𝑆 = ∫ LQED 𝑑4 𝑥 where (107) 1 / − 𝑐2 𝑚) 𝜓 − LQED = 𝜓 (𝑖ℏ𝑐D 𝐹 𝐹𝜇𝜈 and 4𝜇0 𝜇𝜈 / 𝜇 = 𝛾𝜇 (∂𝜇 − 𝑖𝑒𝐴 𝜇 ) D The first, matter term in the Lagrangian leads to the Dirac equation: it describes how elementary, charged, spin 1/2 particles are moved by electromagnetic fields. The second, radiation term leads to Maxwell’s equations, and describes how electromagnetic fields are moved by the charged particle wave function. Together with a few calculating tricks, these equations describe what is usually called quantum electrodynamics, or QED for short. As far as is known today, the relativistic description of the motion of charged mat- Motion Mountain – The Adventure of Physics ter and electromagnetic fields given the QED Lagrangian (107) is perfect: no differences between theory and experiment have ever been found, despite intensive searches and despite a high reward for anybody who would find one. All known predictions fully agree with all measurements. In the most spectacular cases, the correspondence between the- ory and measurement extends to more than thirteen digits. But even more interesting than the precision of QED are certain of its features that are missing in classical electro- dynamics. Let’s have a quick tour. Get ting a first feeling for the Dirac equation copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net The QED Lagrangian implies that the wave function of a charged particle in a potential follows the Dirac equation: 𝑖ℏ𝛾𝜇 (∂𝜇 − 𝑖𝑒𝐴 𝜇 )𝜓 = 𝑚𝑐𝜓 . (108) The many indices should not make us forget that this equation simply states that the eigenvalue of the energy–momentum operator is the rest mass (times the speed of light 𝑐). In other words, the equation states that the wave 𝜓 moves with a phase velocity 𝑐. The wave function 𝜓 has four complex components. Two describe the motion of particles, and two the motion of antiparticles. Each type of particle needs two complex components, because the equation describes spin and particle density. Spin is a rotation, and a rotation requires three real parameters. Spin and density thus require four real parameters; they can be combined into two complex numbers, both for particles and for antiparticles. Each of the four components of the wave function of a relativistic spinning particle Challenge 152 e follows the relativistic Schrödinger–Klein–Gordon equation. This means that the relativ- istic energy–momentum relation is followed by each component separately. The relativistic wave function 𝜓 has the important property that a rotation by 2π Challenge 153 e changes its sign. Only a rotation by 4π leaves the wave function unchanged. This is the typical behaviour of spin 1/2 particles. For this reason, the four-component wave func- tion of a spin 1/2 particle is called a spinor. 192 8 colours and more F I G U R E 84 Klein’s paradox: the motion of a relativistic wave Motion Mountain – The Adventure of Physics function that encounters a very steep potential. Part of the wave function is transmitted; this part is antimatter, as the larger lower component shows. (QuickTime ﬁlm © Bernd Thaller) Antimat ter copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ‘Antimatter’ is now a household term. Interestingly, the concept appeared before there was any experimental evidence for it. The relativistic expression for the energy 𝐸 of an electron with charge 𝑒 in the field of a charge 𝑄 is 𝑄𝑒 𝐸+ = √𝑚2 𝑐4 + 𝑝2 𝑐2 . (109) 4π𝜀0 𝑟 This expression also allows solutions with negative energy and opposite charge −𝑒, if the negative root is used. Quantum theory shows that this is a general property, and these solutions correspond to what is called antimatter. Indeed, the antimatter companion of the electron was predicted in the 1920s by Paul Dirac from his equation (108), which is based on the above relativistic energy relation (109). Unaware of this prediction, Carl Anderson discovered the antielectron in 1932, and called it the positron. (The correct name would have been ‘positon’, without the ‘r’. This correct form is used in the French language.) Anderson was studying cosmic rays, and noticed that some ‘electrons’ were turning the wrong way in the magnetic field he had applied to his apparatus. He checked his apparatus thoroughly, and finally deduced that he had found a particle with the same mass as the electron but with positive electric charge. The existence of positrons has many strange implications. Already in 1928, before their discovery, the Swedish theorist Oskar Klein had pointed out that Dirac’s equation for 8 colours and more 193 electrons makes a strange prediction: when an electron hits a sufficiently steep potential wall, the reflection coefficient is larger than unity. Such a wall will reflect more than is thrown at it. In addition, a large part of the wave function is transmitted through the wall. In 1935, after the discovery of the positron, Werner Heisenberg and Hans Euler Ref. 140 explained the paradox. They found that the Dirac equation predicts that whenever an electric field exceeds the critical value of 𝑚e 𝑐2 𝑚2e 𝑐3 𝐸c = = = 1.3 EV/m , (110) 𝑒𝜆 e 𝑒ℏ the vacuum will spontaneously generate electron–positron pairs, which are then separ- ated by the field. As a result, the original field is reduced. This so-called vacuum polariza- tion is the reason for the reflection coefficient greater than unity found by Klein. Indeed, steep potentials correspond to high electric fields. Vacuum polarization shows that, in contrast to everyday life, the number of particles Motion Mountain – The Adventure of Physics is not a constant in the microscopic domain. Only the difference between particle number and antiparticle number turns out to be conserved. Vacuum polarization thus limits our possibility to count particles in nature! Vacuum polarization is a weak effect. It has been only observed in particle collisions of high energy. In those case, the effect even increases the fine-structure constant! Later Vol. V, page 153 on we will describe truly gigantic examples of vacuum polarization that are postulated around charged black holes. Of course, the generation of electron–positron pairs is not a creation out of nothing, but a transformation of energy into matter. Such processes are part of every relativistic description of nature. Unfortunately, physicists have a habit of calling this transformation copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ‘pair creation’, thus confusing the issue somewhat. The transformation is described by quantum field theory, which we will explore in the next volume. Virtual particles Despite what was said so far, action values smaller than the smallest action value do have a role to play. We have already encountered one example: in a collision between Page 64 two electrons, there is an exchange of virtual photons. We learned that the exchanged virtual photon cannot be observed. Indeed, the action 𝑆 for this exchange obeys 𝑆⩽ℏ. (111) In short, virtual particles appear only as mediators in interactions. They cannot be ob- served. Virtual particles, in contrast to ordinary, real particles, do not obey the relation 𝐸2 − 𝑝2 𝑐2 = 𝑚2 𝑐4 . For example, the kinetic energy can be negative. Indeed, virtual particles are the opposite of ‘free’ or real particles. They may be observed in a vacuum if the measurement time is very short. They are intrinsically short-lived. Virtual photons are the cause for electrostatic potentials, for magnetic fields, for the Casimir effect, for spontaneous emission, for the van der Waals force, and for the Lamb shift in atoms. A more detailed treatment shows that in every situation with vir- tual photons there are also, with even lower probability, virtual electrons and virtual 194 8 colours and more positrons. Massive virtual particles are essential for vacuum polarization, for the limit in the number of the elements, for black-hole radiation and for Unruh radiation. Massive vir- tual particles also play a role in the strong interaction, where they hold the nucleons together in nuclei, and in weak nuclear interaction, where they explain why beta decay happens and why the Sun shines. In particular, virtual particle–antiparticle pairs of matter and virtual radiation particles together form what we call the vacuum. In addition, virtual radiation particles form what are usually called static fields. Virtual particles are needed for a full descrip- tion of all interactions. In particular, virtual particles are responsible for every decay process. Curiosities and fun challenges ab ou t colour and atoms Where is the sea bluest? Sea water, like fresh water, is blue because it absorbs red and green light. The absorption is due to a vibrational band of the water molecule that is due Motion Mountain – The Adventure of Physics Ref. 141 to a combination of symmetric and asymmetric molecular stretches. The absorption is weak, but noticeable. At 700 nm (red), the 1/𝑒 absorption length of water is 1 m. Sea water can also be of bright colour if the sea floor reflects light. In addition, sea wa- ter can be green, if it contains small particles that scatter or absorb blue light. Most often, these particles are soil or plankton. (Satellites can determine plankton content from the ‘greenness’ of the sea.) Thus the sea is especially blue if it is deep, quiet and cold; in that case, the ground is distant, soil is not mixed into the water, and the plankton content is low. The Sargasso Sea is 5 km deep, quiet and cold for most of the year. It is often called the bluest of the Earth’s waters. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Lakes can also be blue if they contain small mineral particles. The particles scatter light and lead to a blue colour for reasons similar to the blue colour of the sky. Such blue lakes are found in many places on Earth. ∗∗ On modern high-precision measurements of the hydrogen spectra, listen to the undis- puted master of the field: enjoy the 2012 talk by Theodor Hänsch, who has devoted a large part of his life to the topic, at www.mediatheque.lindau-nobel.org. ∗∗ The hydrogen atom bears many fascinating aspects. In 2015, Friedmann and Hagen Ref. 142 showed that a well-known formula for π = 3.14159265... can be extracted from the col- our spectrum. Quantum mechanics and colours are beautiful subjects indeed. ∗∗ If atoms contain orbiting electrons, the rotation of the Earth, via the Coriolis accelera- Ref. 136 tion, should have an effect on their motion, and thus on the colour of atoms. This beau- tiful prediction is due to Mark Silverman; the effect is so small, however, that it has not yet been measured. ∗∗ 8 colours and more 195 Light is diffracted by material gratings. Can matter be diffracted by light gratings? Sur- prisingly, it actually can, as predicted by Dirac and Kapitza in 1937. This was accom- Ref. 143 plished for the first time in 1986, using atoms. For free electrons, the feat is more difficult; the clearest confirmation came in 2001, when new laser technology was used to perform a beautiful measurement of the typical diffraction maxima for electrons diffracted by a light grating. ∗∗ Light is totally reflected when it is directed to a dense material at a large enough angle so that it cannot enter the material. A group of Russian physicists have shown that if the Ref. 136 dense material is excited, the intensity of the totally-reflected beam can be amplified. It is unclear whether this will ever lead to applications. ∗∗ The ways people handle single atoms with electromagnetic fields provide many beautiful Motion Mountain – The Adventure of Physics examples of modern applied technologies. Nowadays it is possible to levitate, to trap, to Vol. I, page 344 excite, to photograph, to deexcite and to move single atoms just by shining light onto Ref. 144 them. In 1997, the Nobel Prize in Physics has been awarded to the originators of the field, Steven Chu, Claude Cohen-Tannoudji and William Philips. ∗∗ Ref. 145 Given two mirrors and a few photons, it is possible to capture an atom and keep it floating between the two mirrors. This feat, one of several ways to isolate single atoms, is now Challenge 154 s standard practice in laboratories. Can you imagine how it is done? copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ∗∗ An example of modern research is the study of hollow atoms, i.e., atoms missing a num- ber of inner electrons. They have been discovered in 1990 by J.P. Briand and his group. They appear when a completely ionized atom, i.e., one without any electrons, is brought in contact with a metal. The acquired electrons then orbit on the outside, leaving the inner shells empty, in stark contrast with usual atoms. Such hollow atoms can also be Ref. 146 formed by intense laser irradiation. ∗∗ Relativistic quantum effects can be seen with the unaided eye. The two most important ones concern gold and mercury. The yellow colour of gold – which has atomic number 79 – is due to the transition energy between 5d and 6s electrons, which absorbs blue light. Without relativistic effects, this transition would lie in the ultraviolet, similar to the transition between 4d and 5s electrons for silver, and gold would be colourless. The yellow colour of gold is thus a relativistic effect. Mercury – which has atomic number 80 – has a filled 6s shell. Due to the same re- lativistic effects that appear in gold, these shells are contracted and do not like to form bonds. For this reason, mercury is still liquid a room temperature, in contrast to all other metals. Relativity is thus the reason that mercury is liquid, and that thermometers work. ∗∗ 196 8 colours and more Challenge 155 s Is phosphorus phosphorescent? ∗∗ It is possible to detect the passage of a single photon through an apparatus without ab- Challenge 156 ny sorbing it. How would you do this? Material properties Like the size of hydrogen atoms, also the size of all other atoms is fixed by the quantum of action. Indeed, the quantum of action determines to a large degree the interactions among electrons. By doing so, the quantum of change determines all the interactions between atoms in everyday matter; therefore it determines all other material properties. The elasticity, the plasticity, the brittleness, the magnetic and electric properties of ma- terials are equally fixed by the quantum of action. Only ℏ makes electronics possible! We will study some examples of material properties in the next volume. Various details of the general connection between ℏ and material properties are still a subject of research, Motion Mountain – The Adventure of Physics though none is in contradiction with the quantum of action. Material research is among the most important fields of modern science, and most advances in our standard of living result from it. We will explore some aspects in the next volume. In summary, materials science has confirmed that quantum physics is also the correct description of all materials; quantum physics has confirmed that all material properties of everyday life are of electromagnetic origin; and quantum physics has confirmed that all material properties of everyday life are due to interactions that involve electrons. A tough challenge: the strength of electromagnetism copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net The great physicist Wolfgang Pauli used to say that after his death, the first thing he would ask god would be to explain Sommerfeld’s fine-structure constant. (Others used to comment that after god will have explained it to him, he will think a little, and then snap: ‘Wrong!’) The fine-structure constant, introduced by Arnold Sommerfeld, is the dimensionless Ref. 147 constant of nature whose value is measured to be 𝑒2 1 𝛼= ≈ ≈ 0.007 297 352 5376(50) . (112) 4π𝜀0 ℏ𝑐 137.035 999 679(94) This number first appeared in explanations of the fine structure of atomic colour spectra; Ref. 148 hence its strange name. Sommerfeld was the first to understand its general importance. It is central to quantum electrodynamics for several reasons. First of all, the fine-structure constant describes the strength of electromagnetism. The number 𝛼 results from the interaction of two electric charges 𝑒. Writing Coulomb’s relation for the force 𝐹 between two electrons as ℏ𝑐 𝐹=𝛼 (113) 𝑟2 it becomes clear that the fine-structure constant describes the strength of electromagnet- 8 colours and more 197 ism. A higher value for the fine-structure constant 𝛼 would mean a stronger attraction or repulsion between charged bodies. Thus the value of 𝛼 determines the sizes of atoms, and indeed of all things, as well as all colours in nature. Secondly, it is only because the fine-structure constant 𝛼 is so small that we are able to talk about particles at all. Indeed, only because the fine-structure constant is much smaller than 1 it is possible to distinguish particles from each other. If the number 𝛼 were near to or larger than 1, particles would interact so strongly that it would not be possible to observe them separately or to talk about particles at all. This leads on to the third reason for the importance of the fine-structure constant. Since it is a dimensionless number, it implies some yet-unknown mechanism that fixes its value. Uncovering this mechanism is one of the challenges remaining in our adven- ture. As long as the mechanism remains unknown – as was the case in 2016 – we do not understand the colour and size of a single thing around us! Small changes in the strength of electromagnetic attraction between electrons and protons would have numerous important consequences. Can you describe what would Motion Mountain – The Adventure of Physics happen to the size of people, to the colour of objects, to the colour of the Sun, or to the workings of computers, if the strength were to double? And what if it were to gradually Challenge 157 s drop to half its usual value? Since the 1920s, explaining the value of 𝛼 has been seen as one of the toughest chal- lenges facing modern physics. That is the reason for Pauli’s fantasy. In 1946, during his Nobel Prize lecture, he repeated the statement that a theory that does not determine Ref. 149 this number cannot be complete. Since that time, physicists seem to have fallen into two classes: those who did not dare to take on the challenge, and those who had no clue. This fascinating story still awaits us. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net The problem of the fine-structure constant is so deep that it leads many astray. For example, it is sometimes claimed that it is impossible to change physical units in such a way that ℏ, 𝑐 and 𝑒 are all equal to 1 at the same time, because to do so would change the Challenge 158 s number 𝛼 = 1/137.036...(1). Can you explain why the argument is wrong? A summary on colours and materials In summary, the quantum of action ℏ – together with the interaction between electro- magnetic fields and the electrons inside atoms, molecules, liquids and solids – determ- ines the size, the shape, the colour and the material properties of all things around us. The quantum of action determines mechanical properties such as hardness or elasticity, magnetic properties, thermal properties such as heat capacity or heat of condensation, optical properties such as transparency, and electrical properties such as metallic shine. In addition, the quantum of action determines all chemical and biological aspects of matter. This connection is the topic of the next volume. The strength of the electromagnetic interaction is described by the fine-structure con- stant 𝛼 ≈ 1/137.036. Its value is yet unexplained. Chapter 9 QUA N T UM PH YSIC S I N A N U T SH E L L C ompared to classical physics, quantum theory is definitely more omplex. The basic idea however, is simple: in nature there is a smallest hange, or a smallest action with the value ℏ = 1.1 ⋅ 10−34 Js. More precisely, all of quantum theory can be resumed in one sentence: Motion Mountain – The Adventure of Physics ⊳ In nature, actions or changes smaller than ℏ = 1.054 571 800(13) ⋅ 10−34 Js are not observed. This smallest action value, the quantum of action, leads to all the strange observations made in the microscopic domain, such as the wave behaviour of matter, indeterminacy relations, decoherence, randomness in measurements, indistinguishability, quantization of angular momentum, tunnelling, pair creation, decay and particle reactions. The essence of quantum theory is thus the lack of infinitely small change. The math- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ematics of quantum theory is abstract and involved, though. Was this part of our walk worth the effort? It was: the results are profound, and the accuracy of the description is complete. We first give an overview of these results and then turn to the questions that are still left open. Physical results of quantum theory The existence of a smallest action value in nature leads directly to the main lesson we learned about motion in the quantum part of our adventure: ⊳ If it moves, it is made of quantons, or quantum particles. This statement applies to every physical system, thus to all objects and to all images, i.e., to all matter and radiation. Moving stuff is made of quantons. Stones, water waves, light, sound waves, earthquakes, gelatine and everything else we can interact with is made of quantum particles. In our exploration of relativity we discovered that also horizons and the vacuum can move. If all moving entities are made of quantum particles, what does this imply for horizons and empty space? We can argue that no fundamental problems are expected for horizons, because one way to describe horizons is as an extreme state of matter. But the details for the quantum aspects of vacuum are not simple; they will be the topic of the last part of this adventure. 9 quantum physics in a nutshell 199 Vol. II, page 293 Earlier in our adventure we asked: what are matter, radiation and interactions? Now Vol. II, page 322 we know: they all are composites of elementary quantum particles. In particular, inter- actions are exchanges of elementary quantum particles. An elementary quantum particle is a countable entity that is smaller than its own Compton wavelength. All elementary particles are described by energy–momentum, mass, spin, C, P and T parity. However, as we will see in the next volume, this is not yet the complete list of particle properties. About the intrinsic properties of quantum particles, i.e., those that do not depend on the observer, quantum theory makes a simple statement: ⊳ In nature, all intrinsic properties of quantons, or quantum particles – with the exception of mass – such as spin, electric charge, strong charge, parities Page 128 etc., appear as integer multiples of a basic unit. Since all physical systems are made of quantons, in composed systems all intrinsic properties – with the exception of mass* – either add or multiply. Motion Mountain – The Adventure of Physics In summary, all moving entities are made of quantum particles described by discrete in- trinsic properties. To see how deep this result is, you can apply it to all those moving entities for which it is usually forgotten, such as ghosts, spirits, angels, nymphs, dae- mons, devils, gods, goddesses and souls. You can check yourself what happens when Challenge 159 e their particle nature is taken into account. “ ” Deorum injuriae diis curae.** Tiberius, as reported by Tacitus. R esults on the motion of quantum particles copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Quantons, or quantum particles, differ from everyday particles: quantum particles in- terfere: they behave like a mixture of particles and waves. This property follows directly from the existence of ℏ, the smallest possible action in nature. From the existence of ℏ, quantum theory deduces all its statements about quantum particle motion. We summar- ize the main ones. There is no rest in nature. All objects obey the indeterminacy relation, which states that the indeterminacies in position 𝑥 and momentum 𝑝 follow Δ𝑥Δ𝑝 ⩾ ℏ/2 with ℏ = 1.1 ⋅ 10−34 Js (114) and making rest an impossibility. The state of quantum particles is defined by the same observables as in classical physics, with the difference that observables do not commute. Classical physics appears in the limit that the Planck constant ℏ can effectively be set to zero. Quantum theory introduces a probabilistic element into motion. Probabilities result from the quantum of action through the interactions with the baths that are part of the * More precisely, together with mass, also mixing angles are not quantized. These properties are defined in the next volume. ** ‘Offenses of gods are care of the gods.’ 200 9 quantum physics in a nutshell environment of every physical system. Equivalently, probabilities result in every experi- ment that tries to induce a change that is smaller than the quantum of action. Quantum particles behave like waves. The associated de Broglie wavelength 𝜆 is given by the momentum 𝑝 through ℎ 2πℏ 𝜆= = (115) 𝑝 𝑝 both in the case of matter and of radiation. This relation is the origin of the wave beha- viour of light and matter. The light particles are called photons; their observation is now standard practice. Quantum theory states that particle waves, like all waves, interfere, re- fract, disperse, dampen, can be dampened and can be polarized. This applies to photons, electrons, atoms and molecules. All waves being made of quantum particles, all waves can be seen, touched and moved. Light for example, can be ‘seen’ in photon-photon scattering in vacuum at high energies, can be ‘touched’ using the Compton effect, and can be ‘moved’ by gravitational bending. Matter particles, such as molecules or atoms, Motion Mountain – The Adventure of Physics can be seen in electron microscopes and can be touched and moved with atomic force microscopes. The interference and diffraction of wave particles is observed daily in the electron microscope. Matter waves can be imagined as clouds that rotate locally. In the limit of negligible cloud size, quantum particles can be imagined as rotating little arrows. Equivalently, quantons have a phase. Particles cannot be enclosed forever. Even though matter is impenetrable, quantum theory shows that tight boxes or insurmountable obstacles do not exist. Enclosure is never forever. Waiting long enough always allows us to overcome any boundary, since there is a finite probability to overcome any obstacle. This process is called tunnelling copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net when seen from the spatial point of view and is called decay when seen from the temporal point of view. Tunnelling explains the working of television tubes as well as radioactive decay. All particles and all particle beams can be rotated. Particles possess an intrinsic an- gular momentum called spin, specifying their behaviour under rotations. Bosons have integer spin, fermions have half integer spin. An even number of bound fermions or any number of bound bosons yield a composite boson; an odd number of bound fermions yield a low-energy fermion. Solids are impenetrable because of the fermion character of its electrons in the atoms. Identical particles are indistinguishable. Radiation is made of indistinguishable particles called bosons, matter of fermions. Under exchange of two fermions at space-like separations, the wave function changes sign, whereas for two bosons the wave function remains unchanged. All other properties of quantum particles are the same as for classical particles, namely countability, interaction, mass, charge, angular momentum, energy, momentum, position, as well as impenetrability for matter and penetrability for radiation. Perfect copying machines do not exist. In collisions, particles interact locally, through the exchange of other particles. When matter particles collide, they interact through the exchange of virtual bosons, i.e., off- shell bosons. Motion change is thus due to particle exchange. Exchange bosons of even spin mediate only attractive interactions. Exchange bosons of odd spin mediate repulsive interactions as well. 9 quantum physics in a nutshell 201 The properties of collisions imply the non-conservation of particle number. In col- lisions, particles can appear – i.e., can be ‘created’ – or disappear – i.e., can be ‘annihilated’. This is valid both for bosons and for fermions. The properties of collisions imply the existence of antiparticles, which are regularly observed in experiments. Elementary fermions, in contrast to many elementary bosons, differ from their antiparticles; they can be created and annihilated only in pairs. Element- ary fermions have non-vanishing mass and move slower than light. Particles can decay and be transformed. Detailed investigations show that collisions imply the non-conservation of particle type. In collisions, selected particles can change their intrinsic properties. This observation will be detailed in the next volume. Equival- ently, the quantum of action implies that things break and living beings die. Images, made of radiation, are described by the same observables as matter: position, phase, speed, mass, momentum etc. – though their values and relations differ. Images can only be localized with a precision of the wavelength 𝜆 of the radiation producing them. Motion Mountain – The Adventure of Physics The appearance of Planck’s constant ℏ implies that length scales and time scales exist in nature. Quantum theory introduces a fundamental jitter in every example of motion. Thus the infinitely small is eliminated. In this way, lower limits to structural dimensions and to many other measurable quantities appear. In particular, quantum theory shows that it is impossible that on the electrons in an atom small creatures live in the same way that humans live on the Earth circling the Sun. Quantum theory shows the impossibility of Lilliput. Clocks and metre bars have finite precision, due to the existence of a smallest action and due to their interactions with baths. On the other hand, all measurement apparatuses copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net must contain baths, since otherwise they would not be able to record results. Ref. 150 Quantum effects leave no room for cold fusion, astrology, teleportation, telekinesis, supernatural phenomena, creation out of nothing, multiple universes, or faster than light Page 153 phenomena – the EPR paradox notwithstanding. Achievements in accuracy and precision Apart from the conceptual changes, quantum theory improved the accuracy of predic- tions from the few – if any – digits common in classical mechanics to the full number of digits – sometimes thirteen – that can be measured today. The limited precision is usu- ally not given by the inaccuracy of theory, it is given by the measurement accuracy. In other words, the agreement is only limited by the amount of money the experimenter is willing to spend. Table 8 shows this in more detail. TA B L E 8 Selected comparisons between classical physics, quantum theory and experiment. O b s e r va b l e Clas - Prediction of Measurement Cost sical qua nt u m esti- predic - theory𝑎 m at e tion Simple motion of bodies Indeterminacy 0 Δ𝑥Δ𝑝 ⩾ ℏ/2 (1 ± 10−2 ) ℏ/2 10 k€ 202 9 quantum physics in a nutshell O b s e r va b l e Clas - Prediction of Measurement Cost sical qua nt u m esti- predic - theory𝑎 m at e tion Matter wavelength none 𝜆𝑝 = 2πℏ (1 ± 10−2 ) ℏ 10 k€ Compton wavelength none 𝜆 c = ℎ/𝑚e 𝑐 (1 ± 10−3 ) 𝜆 20 k€ Pair creation rate 0 𝜎𝐸 agrees 100 k€ Radiative decay time in none 𝜏 ∼ 1/𝑛3 (1 ± 10−2 ) 5 k€ hydrogen Smallest angular 0 ℏ/2 (1 ± 10−6 ) ℏ/2 10 k€ momentum Casimir effect/pressure 0 𝑝 = (π2 ℏ𝑐)/(240𝑟4 ) (1 ± 10−3 ) 30 k€ Colours of objects Spectrum of hot objects diverges 𝜆 max = ℎ𝑐/(4.956 𝑘𝑇) (1 ± 10−4 ) Δ𝜆 10 k€ Motion Mountain – The Adventure of Physics Lamb shift none Δ𝜆 = 1057.86(1) MHz (1 ± 10−6 ) Δ𝜆 50 k€ Rydberg constant none 𝑅∞ = 𝑚e 𝑐𝛼2 /2ℎ (1 ± 10−9 ) 𝑅∞ 50 k€ Stefan–Boltzmann none 𝜎 = π2 𝑘4 /60ℏ3 𝑐2 (1 ± 3 ⋅ 10−8 ) 𝜎 20 k€ constant Wien’s displacement none 𝑏 = 𝜆 max 𝑇 (1 ± 10−5 ) 𝑏 20 k€ constant Refractive index of water none 1.34 within a few % 1 k€ Photon-photon scattering 0 from QED: finite agrees 50 M€ Electron gyromagnetic 1 or 2 2.002 319 304 365(7) 2.002 319 304 30 M€ copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ratio 361 53(53) Muon anomalous 0 11 659 1827(63) ⋅ 10−11 11 659 2080(60) ⋅ 10−11 100 M€ magnetic moment Composite matter properties Atom lifetime ≈ 1 μs ∞ > 1020 a 1€ Muonium hyperfine none 4 463 302 542(620) Hz 4 463 302 765(53) Hz 1 M€ splitting Molecular size and shape none from QED within 10−3 20 k€ Page 214 𝑎. All these predictions are calculated from the basic physical constants given in Appendix A. We notice that the predicted values do not differ from the measured ones. If we remember that classical physics does not allow us to calculate any of the measured values, we get an idea of the progress quantum physics has achieved. This advance in understanding is due to the introduction of the quantum of action ℏ. Equivalently, we can state: no description of nature without the quantum of action is complete. In summary, quantum theory is precise and accurate. In the microscopic domain quantum theory is in perfect correspondence with nature; despite prospects of fame and riches, despite the largest number of researchers ever, no contradiction between obser- vation and theory has been found yet. On the other hand, explaining the measured value 9 quantum physics in a nutshell 203 of the fine-structure constant, 𝛼 = 1/137.035 999 074(44), remains an open problem of the electromagnetic interaction. Is quantum theory magic? Studying nature is like experiencing magic. Nature often looks different from what it is. During magic we are fooled – but only if we forget our own limitations. Once we start to see ourselves as part of the game, we start to understand the tricks. That is the fun of magic. The same happens in quantum motion. ∗∗ Nature seems irreversible, even though it isn’t. We never remember the future. We are fooled because we are macroscopic. ∗∗ Nature seems decoherent, even though it isn’t. We are fooled again because we are mac- Motion Mountain – The Adventure of Physics roscopic. ∗∗ There are no clocks in nature. We are fooled by those of everyday life because we are surrounded by a huge number of particles. ∗∗ Motion often seems to disappear, even though it is eternal. We are fooled again, because our senses cannot experience the microscopic domain. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ∗∗ Objects seem distinguishable, even though the statistical properties of their components show that they are not. We are fooled because we live at low energies. ∗∗ Matter seems continuous, even though it isn’t. We are fooled because of the limitations of our senses. ∗∗ Motion seems deterministic in the classical sense, even though it is random. We are fooled again because we are macroscopic. ∗∗ In short, our human condition permanently fools us. The answer to the title question is: classical physics is like magic, and the tricks are uncovered by quantum theory. That is its main attraction. Q uantum theory is exact, bu t can d o more We can summarize this part of our adventure with a simple statement: 204 9 quantum physics in a nutshell ⊳ Quantum physics is the description of matter and radiation without the concept of infinitely small. All change in nature, in fact, everything is described by finite quantities, and above all, by the smallest change possible in nature, the quantum of action ℏ = 1.054 571 800(13) ⋅ 10−34 Js. All experiments, without exception, show that the quantum of action ℏ is the smallest observable change. The description of nature with the quantum of action is thus exact and final. The smallest measurable action ℏ, like the maximum energy speed 𝑐, is a fun- damental property of nature. One could also call both of them fundamental truths. Since quantum theory follows logically and completely from the smallest measurable action ℏ, the simplest way – and the only way – to disprove quantum theory is to find an Challenge 160 e observation that contradicts the smallest change value ℏ. Try it! Even though we have deduced a fundamental property of nature, if we turn back to the Page 15 start of our exploration of quantum theory, we cannot hide a certain disappointment. We Motion Mountain – The Adventure of Physics know that classical physics cannot explain life. Searching for the details of microscopic motion, we encountered so many interesting aspects that we have not yet achieved the explanation of life. For example, we know what determines the speed of electrons in atoms, but we do not know what determines the running speed of an athlete. In fact, we have not even discussed the properties of any solid or liquid, let alone those of more complex structures like living beings. In other terms, after this introduction into quantum theory, we must still connect quantum processes to our everyday world. Therefore, the topic of the next volume will be the exploration of the motion of and inside living things – and of the motion inside all copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net kind of matter, from solids to stars, using the quantum of action as a foundation. After that, we will explore the motion of empty space. Appendix A U N I T S , M E A SU R E M E N T S A N D C ON STA N T S M easurements are comparisons with standards. Standards are based on units. any different systems of units have been used throughout the world. ost of these standards confer power to the organization in charge of them. Such power can be misused; this is the case today, for example in the computer in- Motion Mountain – The Adventure of Physics dustry, and was so in the distant past. The solution is the same in both cases: organize an independent and global standard. For measurement units, this happened in the eighteenth century: in order to avoid misuse by authoritarian institutions, to eliminate problems with differing, changing and irreproducible standards, and – this is not a joke – to simplify tax collection and to make it more just, a group of scientists, politicians and economists agreed on a set of units. It is called the Système International d’Unités, abbreviated SI, and is defined by an international treaty, the ‘Convention du Mètre’. The units are maintained by an international organization, the ‘Conférence Générale des Poids et Mesures’, and its daughter organizations, the ‘Commission Internationale des Poids et Mesures’ and the ‘Bureau International des Poids et Mesures’ (BIPM). All copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Ref. 151 originated in the times just before the French revolution. SI units All SI units are built from seven base units. Their simplest definitions, translated from French into English, are the following ones, together with the dates of their formulation and a few comments: ‘The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.’ (1967) The 2019 definition is equivalent, but much less clear.* ‘The metre is the length of the path travelled by light in vacuum during a time inter- val of 1/299 792 458 of a second.’ (1983) The 2019 definition is equivalent, but much less clear.* ‘The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 15 ⋅ 10−34 when expressed in the unit J ⋅ s, which is equal to kg ⋅ m2 ⋅ s−1 .’ (2019)* ‘The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602 176 634 ⋅ 10−19 when ex- pressed in the unit C, which is equal to A ⋅ s.’ (2019)* This definition is equivalent to: One ampere is 6.241 509 074... ⋅ 1018 elementary charges per second. ‘The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by 206 a units, measurements and constants taking the fixed numerical value of the Boltzmann constant 𝑘 to be 1.380649 ⋅10−23 when expressed in the unit J ⋅ K−1 .’ (2019)* ‘The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.02214076 ⋅ 1023 elementary entities.’ (2019)* ‘The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 ⋅ 1012 hertz and has a radiant intensity in that direction of (1/683) watt per steradian.’ (1979) The 2019 definition is equivalent, but much less clear.* We note that both time and length units are defined as certain properties of a standard example of motion, namely light. In other words, also the Conférence Générale des Poids et Mesures makes the point that the observation of motion is a prerequisite for the defin- ition and construction of time and space. Motion is the fundament of every observation and of all measurement. By the way, the use of light in the definitions had been proposed already in 1827 by Jacques Babinet.** Motion Mountain – The Adventure of Physics From these basic units, all other units are defined by multiplication and division. Thus, all SI units have the following properties: SI units form a system with state-of-the-art precision: all units are defined with a pre- cision that is higher than the precision of commonly used measurements. Moreover, the precision of the definitions is regularly being improved. The present relative uncertainty of the definition of the second is around 10−14 , for the metre about 10−10 , for the kilo- gram about 10−9 , for the ampere 10−7 , for the mole less than 10−6 , for the kelvin 10−6 and for the candela 10−3 . SI units form an absolute system: all units are defined in such a way that they can copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net be reproduced in every suitably equipped laboratory, independently, and with high pre- cision. This avoids as much as possible any error or misuse by the standard-setting or- ganization. In fact, the SI units are as now as near as possible to Planck’s natural units, which are presented below. In practice, the SI is now an international standard defining the numerical values of the seven constants Δ𝜈Cs , 𝑐, ℏ, 𝑒, 𝑘, 𝑁A and 𝐾cd . After over 200 years of discussions, the CGPM has little left to do. SI units form a practical system: the base units are quantities of everyday magnitude. Frequently used units have standard names and abbreviations. The complete list includes the seven base units just given, the supplementary units, the derived units and the ad- mitted units. The supplementary SI units are two: the unit for (plane) angle, defined as the ratio of arc length to radius, is the radian (rad). For solid angle, defined as the ratio of the subtended area to the square of the radius, the unit is the steradian (sr). The derived units with special names, in their official English spelling, i.e., without capital letters and accents, are: * The symbols of the seven units are s, m, kg, A, K, mol and cd. The full offical definitions are found at Ref. 152 www.bipm.org. For more details about the levels of the caesium atom, consult a book on atomic physics. The Celsius scale of temperature 𝜃 is defined as: 𝜃/°C = 𝑇/K − 273.15; note the small difference with the number appearing in the definition of the kelvin. In the definition of the candela, the frequency of the light corresponds to 555.5 nm, i.e., green colour, around the wavelength to which the eye is most sensitive. ** Jacques Babinet (1794–1874), French physicist who published important work in optics. a units, measurements and constants 207 Name A bbre v iat i o n Name A b b r e v i at i o n hertz Hz = 1/s newton N = kg m/s2 pascal Pa = N/m2 = kg/m s2 joule J = Nm = kg m2 /s2 watt W = kg m2 /s3 coulomb C = As volt V = kg m2 /As3 farad F = As/V = A2 s4 /kg m2 ohm Ω = V/A = kg m2 /A2 s3 siemens S = 1/Ω weber Wb = Vs = kg m2 /As2 tesla T = Wb/m2 = kg/As2 = kg/Cs henry H = Vs/A = kg m2 /A2 s2 degree Celsius °C (see definition of kelvin) lumen lm = cd sr lux lx = lm/m2 = cd sr/m2 becquerel Bq = 1/s gray Gy = J/kg = m2 /s2 sievert Sv = J/kg = m2 /s2 katal kat = mol/s We note that in all definitions of units, the kilogram only appears to the powers of 1, Motion Mountain – The Adventure of Physics Challenge 161 s 0 and −1. Can you try to formulate the reason? The admitted non-SI units are minute, hour, day (for time), degree 1° = π/180 rad, minute 1 = π/10 800 rad, second 1 = π/648 000 rad (for angles), litre, and tonne. All other units are to be avoided. All SI units are made more practical by the introduction of standard names and ab- breviations for the powers of ten, the so-called prefixes:* Power Name Power Name Power Name Power Name 101 deca da 10−1 deci d 1018 Exa E 10−18 atto a copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 102 hecto h 10−2 centi c 1021 Zetta Z 10−21 zepto z 103 kilo k 10−3 milli m 1024 Yotta Y 10−24 yocto y 106 Mega M 10−6 micro μ unofficial: Ref. 153 109 Giga G 10−9 nano n 1027 Xenta X 10−27 xenno x 1012 Tera T 10−12 pico p 1030 Wekta W 10−30 weko w 1015 Peta P 10−15 femto f 1033 Vendekta V 10−33 vendeko v 1036 Udekta U 10−36 udeko u SI units form a complete system: they cover in a systematic way the full set of ob- servables of physics. Moreover, they fix the units of measurement for all other sciences as well. * Some of these names are invented (yocto to sound similar to Latin octo ‘eight’, zepto to sound similar to Latin septem, yotta and zetta to resemble them, exa and peta to sound like the Greek words ἑξάκις and πεντάκις for ‘six times’ and ‘five times’, the unofficial ones to sound similar to the Greek words for nine, ten, eleven and twelve); some are from Danish/Norwegian (atto from atten ‘eighteen’, femto from femten ‘fifteen’); some are from Latin (from mille ‘thousand’, from centum ‘hundred’, from decem ‘ten’, from nanus ‘dwarf’); some are from Italian (from piccolo ‘small’); some are Greek (micro is from μικρός ‘small’, deca/deka from δέκα ‘ten’, hecto from ἑκατόν ‘hundred’, kilo from χίλιοι ‘thousand’, mega from μέγας ‘large’, giga from γίγας ‘giant’, tera from τέρας ‘monster’). Translate: I was caught in such a traffic jam that I needed a microcentury for a picoparsec and that my Challenge 162 e car’s fuel consumption was two tenths of a square millimetre. 208 a units, measurements and constants SI units form a universal system: they can be used in trade, in industry, in commerce, at home, in education and in research. They could even be used by extraterrestrial civil- izations, if they existed. SI units form a self-consistent system: the product or quotient of two SI units is also an SI unit. This means that in principle, the same abbreviation, e.g. ‘SI’, could be used for every unit. The SI units are not the only possible set that could fulfil all these requirements, but they are the only existing system that does so.* The meaning of measurement Every measurement is a comparison with a standard. Therefore, any measurement re- Challenge 163 e quires matter to realize the standard (even for a speed standard), and radiation to achieve the comparison. The concept of measurement thus assumes that matter and radiation ex- ist and can be clearly separated from each other. Motion Mountain – The Adventure of Physics Every measurement is a comparison. Measuring thus implies that space and time ex- ist, and that they differ from each other. Every measurement produces a measurement result. Therefore, every measurement implies the storage of the result. The process of measurement thus implies that the situ- ation before and after the measurement can be distinguished. In other terms, every meas- urement is an irreversible process. Every measurement is a process. Thus every measurement takes a certain amount of time and a certain amount of space. All these properties of measurements are simple but important. Beware of anybody copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net who denies them. Planck ’ s natural units Since the exact form of many equations depends on the system of units used, theoretical physicists often use unit systems optimized for producing simple equations. The chosen units and the values of the constants of nature are related. In microscopic physics, the system of Planck’s natural units is frequently used. They are defined by setting 𝑐 = 1, ℏ = 1, 𝐺 = 1, 𝑘 = 1, 𝜀0 = 1/4π and 𝜇0 = 4π. Planck units are thus defined from combinations of fundamental constants; those corresponding to the fundamental SI units are given in Table 10.** The table is also useful for converting equations written in natural units back Challenge 164 e to SI units: just substitute every quantity 𝑋 by 𝑋/𝑋Pl. * Apart from international units, there are also provincial units. Most provincial units still in use are of Roman origin. The mile comes from milia passum, which used to be one thousand (double) strides of about 1480 mm each; today a nautical mile, once defined as minute of arc on the Earth’s surface, is defined as exactly 1852 m. The inch comes from uncia/onzia (a twelfth – now of a foot). The pound (from pondere ‘to weigh’) is used as a translation of libra – balance – which is the origin of its abbreviation lb. Even the habit of counting in dozens instead of tens is Roman in origin. These and all other similarly funny units – like the system in which all units start with ‘f’, and which uses furlong/fortnight as its unit of velocity – are now officially defined as multiples of SI units. ** The natural units 𝑥Pl given here are those commonly used today, i.e., those defined using the constant ℏ, and not, as Planck originally did, by using the constant ℎ = 2πℏ. The electromagnetic units can also be defined with other factors than 4π𝜀0 in the expressions: for example, using 4π𝜀0 𝛼, with the fine-structure Page 196 constant 𝛼, gives 𝑞Pl = 𝑒. For the explanation of the numbers between brackets, see below. a units, measurements and constants 209 TA B L E 10 Planck’s (uncorrected) natural units. Name Definition Va l u e Basic units the Planck length 𝑙Pl = √ℏ𝐺/𝑐3 = 1.616 0(12) ⋅ 10−35 m the Planck time 𝑡Pl = √ℏ𝐺/𝑐5 = 5.390 6(40) ⋅ 10−44 s the Planck mass 𝑚Pl = √ℏ𝑐/𝐺 = 21.767(16) μg 6 the Planck current 𝐼Pl = √4π𝜀0 𝑐 /𝐺 = 3.479 3(22) ⋅ 1025 A the Planck temperature 𝑇Pl = √ℏ𝑐5 /𝐺𝑘2 = 1.417 1(91) ⋅ 1032 K Trivial units the Planck velocity 𝑣Pl = 𝑐 = 0.3 Gm/s the Planck angular momentum 𝐿 Pl = ℏ = 1.1 ⋅ 10−34 Js Motion Mountain – The Adventure of Physics the Planck action 𝑆aPl = ℏ = 1.1 ⋅ 10−34 Js the Planck entropy 𝑆ePl = 𝑘 = 13.8 yJ/K Composed units the Planck mass density 𝜌Pl = 𝑐5 /𝐺2 ℏ = 5.2 ⋅ 1096 kg/m3 the Planck energy 𝐸Pl = √ℏ𝑐5 /𝐺 = 2.0 GJ = 1.2 ⋅ 1028 eV the Planck momentum 𝑝Pl = √ℏ𝑐3 /𝐺 = 6.5 Ns 5 the Planck power 𝑃Pl = 𝑐 /𝐺 = 3.6 ⋅ 1052 W the Planck force 𝐹Pl = 𝑐4 /𝐺 = 1.2 ⋅ 1044 N copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net the Planck pressure 𝑝Pl = 𝑐7 /𝐺ℏ = 4.6 ⋅ 10113 Pa the Planck acceleration 𝑎Pl = √𝑐7 /ℏ𝐺 = 5.6 ⋅ 1051 m/s2 the Planck frequency 𝑓Pl = √𝑐5 /ℏ𝐺 = 1.9 ⋅ 1043 Hz the Planck electric charge 𝑞Pl = √4π𝜀0 𝑐ℏ = 1.9 aC = 11.7 e 4 the Planck voltage 𝑈Pl = √𝑐 /4π𝜀0 𝐺 = 1.0 ⋅ 1027 V the Planck resistance 𝑅Pl = 1/4π𝜀0 𝑐 = 30.0 Ω 3 the Planck capacitance 𝐶Pl = 4π𝜀0 √ℏ𝐺/𝑐 = 1.8 ⋅ 10−45 F the Planck inductance 𝐿 Pl = (1/4π𝜀0 )√ℏ𝐺/𝑐7 = 1.6 ⋅ 10−42 H the Planck electric field 𝐸Pl = √𝑐7 /4π𝜀0 ℏ𝐺2 = 6.5 ⋅ 1061 V/m the Planck magnetic flux density 𝐵Pl = √𝑐5 /4π𝜀0 ℏ𝐺2 = 2.2 ⋅ 1053 T The natural units are important for another reason: whenever a quantity is sloppily called ‘infinitely small (or large)’, the correct expression is ‘as small (or as large) as the corres- ponding corrected Planck unit’. As explained throughout the text, and especially in the Vol. VI, page 37 final part, this substitution is possible because almost all Planck units provide, within a correction factor of order 1, the extremal value for the corresponding observable – some an upper and some a lower limit. Unfortunately, these correction factors are not yet widely known. The exact extremal value for each observable in nature is obtained 210 a units, measurements and constants when 𝐺 is substituted by 4𝐺 and 4π𝜀0 by 4π𝜀0 𝛼 in all Planck quantities. These extremal values, or corrected Planck units, are the true natural units. To exceed the extremal values Challenge 165 s is possible only for some extensive quantities. (Can you find out which ones?) Other unit systems A central aim of research in high-energy physics is the calculation of the strengths of all interactions; therefore it is not practical to set the gravitational constant 𝐺 to unity, as in the Planck system of units. For this reason, high-energy physicists often only set 𝑐 = ℏ = 𝑘 = 1 and 𝜇0 = 1/𝜀0 = 4π,* leaving only the gravitational constant 𝐺 in the equations. In this system, only one fundamental unit exists, but its choice is free. Often a stand- ard length is chosen as the fundamental unit, length being the archetype of a measured quantity. The most important physical observables are then related by 1/[𝑙2 ] = [𝐸]2 = [𝐹] = [𝐵] = [𝐸electric] , Motion Mountain – The Adventure of Physics 1/[𝑙] = [𝐸] = [𝑚] = [𝑝] = [𝑎] = [𝑓] = [𝐼] = [𝑈] = [𝑇] , 1 = [𝑣] = [𝑞] = [𝑒] = [𝑅] = [𝑆action] = [𝑆entropy ] = ℏ = 𝑐 = 𝑘 = [𝛼] , (116) [𝑙] = 1/[𝐸] = [𝑡] = [𝐶] = [𝐿] and [𝑙]2 =1/[𝐸]2 = [𝐺] = [𝑃] where we write [𝑥] for the unit of quantity 𝑥. Using the same unit for time, capacitance and inductance is not to everybody’s taste, however, and therefore electricians do not use this system.** Often, in order to get an impression of the energies needed to observe an effect un- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net der study, a standard energy is chosen as fundamental unit. In particle physics the most common energy unit is the electron volt eV, defined as the kinetic energy acquired by an electron when accelerated by an electrical potential difference of 1 volt (‘proton volt’ would be a better name). Therefore one has 1 eV = 1.6 ⋅ 10−19 J, or roughly 1 1 eV ≈ 6 aJ (117) which is easily remembered. The simplification 𝑐 = ℏ = 1 yields 𝐺 = 6.9 ⋅ 10−57 eV−2 and allows one to use the unit eV also for mass, momentum, temperature, frequency, time Challenge 166 e and length, with the respective correspondences 1 eV ≡ 1.8 ⋅ 10−36 kg ≡ 5.4 ⋅ 10−28 Ns ≡ 242 THz ≡ 11.6 kK and 1 eV−1 ≡ 4.1 fs ≡ 1.2 μm. * Other definitions for the proportionality constants in electrodynamics lead to the Gaussian unit system often used in theoretical calculations, the Heaviside–Lorentz unit system, the electrostatic unit system, and Ref. 154 the electromagnetic unit system, among others. ** In the list, 𝑙 is length, 𝐸 energy, 𝐹 force, 𝐸electric the electric and 𝐵 the magnetic field, 𝑚 mass, 𝑝 momentum, 𝑎 acceleration, 𝑓 frequency, 𝐼 electric current, 𝑈 voltage, 𝑇 temperature, 𝑣 speed, 𝑞 charge, 𝑅 resistance, 𝑃 power, 𝐺 the gravitational constant. The web page www.chemie.fu-berlin.de/chemistry/general/units_en.html provides a tool to convert various units into each other. Researchers in general relativity often use another system, in which the Schwarzschild radius 𝑟s = 2𝐺𝑚/𝑐2 is used to measure masses, by setting 𝑐 = 𝐺 = 1. In this case, mass and length have the same dimension, and ℏ has the dimension of an area. a units, measurements and constants 211 To get some feeling for the unit eV, the following relations are useful. Room temper- ature, usually taken as 20°C or 293 K, corresponds to a kinetic energy per particle of 0.025 eV or 4.0 zJ. The highest particle energy measured so far belongs to a cosmic ray Ref. 155 with an energy of 3 ⋅ 1020 eV or 48 J. Down here on the Earth, an accelerator able to pro- duce an energy of about 105 GeV or 17 nJ for electrons and antielectrons has been built, and one able to produce an energy of 14 TeV or 2.2 μJ for protons will be finished soon. Both are owned by CERN in Geneva and have a circumference of 27 km. The lowest temperature measured up to now is 280 pK, in a system of rhodium Ref. 156 nuclei held inside a special cooling system. The interior of that cryostat may even be the coolest point in the whole universe. The kinetic energy per particle correspond- ing to that temperature is also the smallest ever measured: it corresponds to 24 feV or 3.8 vJ = 3.8 ⋅ 10−33 J. For isolated particles, the record seems to be for neutrons: kinetic energies as low as 10−7 eV have been achieved, corresponding to de Broglie wavelengths of 60 nm. Motion Mountain – The Adventure of Physics Curiosities and fun challenges ab ou t units The Planck length is roughly the de Broglie wavelength 𝜆 B = ℎ/𝑚𝑣 of a man walking Ref. 157 comfortably (𝑚 = 80 kg, 𝑣 = 0.5 m/s); this motion is therefore aptly called the ‘Planck stroll.’ ∗∗ The Planck mass is equal to the mass of about 1019 protons. This is roughly the mass of a human embryo at about ten days of age. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ∗∗ The most precisely measured quantities in nature are the frequencies of certain milli- Ref. 158 second pulsars, the frequency of certain narrow atomic transitions, and the Rydberg constant of atomic hydrogen, which can all be measured as precisely as the second is defined. The caesium transition that defines the second has a finite linewidth that limits the achievable precision: the limit is about 14 digits. ∗∗ The most precise clock ever built, using microwaves, had a stability of 10−16 during a Ref. 159 running time of 500 s. For longer time periods, the record in 1997 was about 10−15 ; but Ref. 160 values around 10−17 seem within technological reach. The precision of clocks is limited for short measuring times by noise, and for long measuring times by drifts, i.e., by sys- tematic effects. The region of highest stability depends on the clock type; it usually lies between 1 ms for optical clocks and 5000 s for masers. Pulsars are the only type of clock for which this region is not known yet; it certainly lies at more than 20 years, the time elapsed at the time of writing since their discovery. ∗∗ The shortest times measured are the lifetimes of certain ‘elementary’ particles. In par- Ref. 161 ticular, the lifetime of certain D mesons have been measured at less than 10−23 s. Such times are measured using a bubble chamber, where the track is photographed. Can you 212 a units, measurements and constants Challenge 167 s estimate how long the track is? (This is a trick question – if your length cannot be ob- served with an optical microscope, you have made a mistake in your calculation.) ∗∗ The longest times encountered in nature are the lifetimes of certain radioisotopes, over 1015 years, and the lower limit of certain proton decays, over 1032 years. These times are thus much larger than the age of the universe, estimated to be fourteen thousand million Ref. 162 years. ∗∗ Variations of quantities are often much easier to measure than their values. For example, in gravitational wave detectors, the sensitivity achieved in 1992 was Δ𝑙/𝑙 = 3 ⋅ 10−19 for Ref. 163 lengths of the order of 1 m. In other words, for a block of about a cubic metre of metal it is possible to measure length changes about 3000 times smaller than a proton radius. These set-ups are now being superseded by ring interferometers. Ring interferometers Motion Mountain – The Adventure of Physics measuring frequency differences of 10−21 have already been built; and they are still being Ref. 164 improved. Precision and accuracy of measurements Measurements are the basis of physics. Every measurement has an error. Errors are due to lack of precision or to lack of accuracy. Precision means how well a result is reproduced when the measurement is repeated; accuracy is the degree to which a measurement cor- responds to the actual value. Lack of precision is due to accidental or random errors; they are best measured by the copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net standard deviation, usually abbreviated 𝜎; it is defined through 1 𝑛 𝜎2 = ∑(𝑥 − 𝑥)̄ 2 , (118) 𝑛 − 1 𝑖=1 𝑖 where 𝑥̄ is the average of the measurements 𝑥𝑖 . (Can you imagine why 𝑛 − 1 is used in Challenge 168 s the formula instead of 𝑛?) For most experiments, the distribution of measurement values tends towards a nor- mal distribution, also called Gaussian distribution, whenever the number of measure- ments is increased. The distribution, shown in Figure 85, is described by the expression (𝑥−𝑥)̄ 2 𝑁(𝑥) ≈ e− 2𝜎2 . (119) The square 𝜎2 of the standard deviation is also called the variance. For a Gaussian distri- Challenge 169 e bution of measurement values, 2.35𝜎 is the full width at half maximum. Lack of accuracy is due to systematic errors; usually these can only be estimated. This estimate is often added to the random errors to produce a total experimental error, some- Ref. 165 times also called total uncertainty. The relative error or uncertainty is the ratio between the error and the measured value. For example, a professional measurement will give a result such as 0.312(6) m. The a units, measurements and constants 213 N number of measurements standard deviation full width at half maximum (FWHM) limit curve for a large number of measurements: the Gaussian distribution x x average value measured values Motion Mountain – The Adventure of Physics F I G U R E 85 A precision experiment and its measurement distribution. The precision is high if the width of the distribution is narrow; the accuracy is high if the centre of the distribution agrees with the actual value. number between the parentheses is the standard deviation 𝜎, in units of the last digits. As above, a Gaussian distribution for the measurement results is assumed. Therefore, a Challenge 170 e value of 0.312(6) m implies that the actual value is expected to lie copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net — within 1𝜎 with 68.3 % probability, thus in this example within 0.312 ± 0.006 m; — within 2𝜎 with 95.4 % probability, thus in this example within 0.312 ± 0.012 m; — within 3𝜎 with 99.73 % probability, thus in this example within 0.312 ± 0.018 m; — within 4𝜎 with 99.9937 % probability, thus in this example within 0.312 ± 0.024 m; — within 5𝜎 with 99.999 943 % probability, thus in this example within 0.312 ± 0.030 m; — within 6𝜎 with 99.999 999 80 % probability, thus within 0.312 ± 0.036 m; — within 7𝜎 with 99.999 999 999 74 % probability, thus within 0.312 ± 0.041 m. Challenge 171 s (Do the latter numbers make sense?) Note that standard deviations have one digit; you must be a world expert to use two, and a fool to use more. If no standard deviation is given, a (1) is assumed. As a result, among professionals, 1 km and 1000 m are not the same length! What happens to the errors when two measured values 𝐴 and 𝐵 are added or subtrac- ted? If the all measurements are independent – or uncorrelated – the standard deviation of the sum and that of difference is given by 𝜎 = √𝜎𝐴2 + 𝜎𝐵2 . For both the product or ratio of two measured and uncorrelated values 𝐶 and 𝐷, the result is 𝜌 = √𝜌𝐶2 + 𝜌𝐷2 , where the 𝜌 terms are the relative standard deviations. Challenge 172 s Assume you measure that an object moves 1 m in 3 s: what is the measured speed value? 214 a units, measurements and constants Limits to precision What are the limits to accuracy and precision? There is no way, even in principle, to measure a length 𝑥 to a precision higher than about 61 digits, because in nature, the ratio between the largest and the smallest measurable length is Δ𝑥/𝑥 > 𝑙Pl/𝑑horizon = 10−61 . Challenge 173 e (Is this ratio valid also for force or for volume?) In the final volume of our text, studies Vol. VI, page 94 of clocks and metre bars strengthen this theoretical limit. But it is not difficult to deduce more stringent practical limits. No imaginable machine can measure quantities with a higher precision than measuring the diameter of the Earth within the smallest length ever measured, about 10−19 m; that is about 26 digits of preci- sion. Using a more realistic limit of a 1000 m sized machine implies a limit of 22 digits. If, as predicted above, time measurements really achieve 17 digits of precision, then they are nearing the practical limit, because apart from size, there is an additional practical restriction: cost. Indeed, an additional digit in measurement precision often means an additional digit in equipment cost. Motion Mountain – The Adventure of Physics Physical constants In physics, general observations are deduced from more fundamental ones. As a con- sequence, many measurements can be deduced from more fundamental ones. The most fundamental measurements are those of the physical constants. The following tables give the world’s best values of the most important physical con- stants and particle properties – in SI units and in a few other common units – as pub- Ref. 166 lished in the standard references. The values are the world averages of the best measure- ments made up to the present. As usual, experimental errors, including both random copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net and estimated systematic errors, are expressed by giving the standard deviation in the last digits. In fact, behind each of the numbers in the following tables there is a long Ref. 167 story which is worth telling, but for which there is not enough room here. In principle, all quantitative properties of matter can be calculated with quantum the- Vol. V, page 261 ory – more precisely, equations of the standard model of particle – and a set of basic physical constants that are given in the next table. For example, the colour, density and elastic properties of any material can be predicted, in principle, in this way. TA B L E 11 Basic physical constants. Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. 𝑎 Constants that define the SI measurement units Vacuum speed of light 𝑐 𝑐 299 792 458 m/s 0 Original Planck constant 𝑐 ℎ 6.626 070 15 ⋅ 10−34 Js 0 Reduced Planck constant, ℏ 1.054 571 817 ... ⋅ 10−34 Js 0 quantum of action Positron charge 𝑐 𝑒 0.160 217 6634 aC 0 Boltzmann constant 𝑐 𝑘 1.380 649 ⋅ 10−23 J/K 0 Avogadro’s number 𝑁A 6.022 140 76 ⋅ 1023 1/mol 0 Constant that should define the SI measurement units a units, measurements and constants 215 TA B L E 11 (Continued) Basic physical constants. Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. 𝑎 Gravitational constant 𝐺 6.674 30(15) ⋅ 10−11 Nm2 /kg2 2.2 ⋅ 10−5 Other fundamental constants Number of space-time dimensions 3+1 0𝑏 2 Fine-structure constant 𝑑 or 𝛼 = 4π𝜀𝑒 ℏ𝑐 1/137.035 999 084(21) 1.5 ⋅ 10−10 0 e.m. coupling constant = 𝑔em (𝑚2e 𝑐2 ) = 0.007 297 352 5693(11) 1.5 ⋅ 10−10 Fermi coupling constant 𝑑 or 𝐺F /(ℏ𝑐)3 1.166 3787(6) ⋅ 10−5 GeV−2 5.1 ⋅ 10−7 weak coupling constant 𝛼w (𝑀Z ) = 𝑔w2 /4π 1/30.1(3) 1 ⋅ 10−2 Strong coupling constant 𝑑 𝛼s (𝑀Z ) = 𝑔s2 /4π 0.1179(10) 8.5 ⋅ 10−3 Weak mixing angle sin2 𝜃W (𝑀𝑆) 0.231 22(4) 1.7 ⋅ 10−4 sin2 𝜃W (on shell) 0.222 90(30) 1.3 ⋅ 10−3 = 1 − (𝑚W /𝑚Z )2 Motion Mountain – The Adventure of Physics 0.97383(24) 0.2272(10) 0.00396(9) CKM quark mixing matrix |𝑉| ( 0.2271(10) 0.97296(24) 0.04221(80) ) 0.00814(64) 0.04161(78) 0.999100(34) Jarlskog invariant 𝐽 3.08(18) ⋅ 10−5 0.82(2) 0.55(4) 0.150(7) PMNS neutrino mixing m. |𝑃| (0.37(13) 0.57(11) 0.71(7) ) 0.41(13) 0.59(10) 0.69(7) Electron mass 𝑚e 9.109 383 7015(28) ⋅ 10−31 kg 3.0 ⋅ 10−10 5.485 799 090 65(16) ⋅ 10−4 u 2.9 ⋅ 10−11 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 0.510 998 950 00(15) MeV 3.0 ⋅ 10−10 Muon mass 𝑚μ 1.883 531 627(42) ⋅ 10−28 kg 2.2 ⋅ 10−8 105.658 3755(23) MeV 2.2 ⋅ 10−8 Tau mass 𝑚𝜏 1.776 82(12) GeV/𝑐2 6.8 ⋅ 10−5 El. neutrino mass 𝑚𝜈e < 2 eV/𝑐2 Muon neutrino mass 𝑚𝜈𝜇 < 2 eV/𝑐2 Tau neutrino mass 𝑚𝜈𝜏 < 2 eV/𝑐2 Up quark mass 𝑢 21.6(+0.49/ − 0.26) MeV/𝑐2 Down quark mass 𝑑 4.67(+0.48/ − 0.17) MeV/𝑐2 Strange quark mass 𝑠 93(+11/ − 5) MeV/𝑐2 Charm quark mass 𝑐 1.27(2) GeV/𝑐2 Bottom quark mass 𝑏 4.18(3) GeV/𝑐2 Top quark mass 𝑡 172.9(0.4) GeV/𝑐2 Photon mass γ < 2 ⋅ 10−54 kg W boson mass 𝑊± 80.379(12) GeV/𝑐2 Z boson mass 𝑍0 91.1876(21) GeV/𝑐2 Higgs mass H 125.10(14) GeV/𝑐2 Gluon mass g1...8 c. 0 MeV/𝑐2 216 a units, measurements and constants 𝑎. Uncertainty: standard deviation of measurement errors. 𝑏. Measured from to 10−19 m to 1026 m. 𝑐. Defining constant. 𝑑. All coupling constants depend on the 4-momentum transfer, as explained in the section on Page 131 renormalization. Fine-structure constant is the traditional name for the electromagnetic coup- ling constant 𝑔em in the case of a 4-momentum transfer of 𝑄2 = 𝑚2e 𝑐2 , which is the smallest one possible. At higher momentum transfers it has larger values, e.g., 𝑔em (𝑄2 = 𝑀W 2 2 𝑐 ) ≈ 1/128. In contrast, the strong coupling constant has lover values at higher momentum transfers; e.g., 𝛼s (34 GeV) = 0.14(2). Why do all these basic constants have the values they have? For any basic constant with a dimension, such as the quantum of action ℏ, the numerical value has only historical meaning. It is 1.054 ⋅ 10−34 Js because of the SI definition of the joule and the second. The question why the value of a dimensional constant is not larger or smaller therefore always requires one to understand the origin of some dimensionless number giving the ratio between the constant and the corresponding natural unit that is defined with 𝑐, 𝐺, Motion Mountain – The Adventure of Physics Page 208 𝑘, 𝑁A and ℏ. Details and values for the natural units are given in the dedicated section. In other words, understanding the sizes of atoms, people, trees and stars, the duration of molecular and atomic processes, or the mass of nuclei and mountains, implies under- standing the ratios between these values and the corresponding natural units. The key to understanding nature is thus the understanding of all measurement ratios, and thus of all dimensionless constants. This quest, including the understanding of the fine-structure constant 𝛼 itself, is completed only in the final volume of our adventure. The basic constants yield the following useful high-precision observations. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net TA B L E 12 Derived physical constants. Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. Vacuum permeability 𝜇0 1.256 637 062 12(19) μH/m 1.5 ⋅ 10−10 Vacuum permittivity 𝜀0 = 1/𝜇0 𝑐2 8.854 187 8128(13) pF/m 1.5 ⋅ 10−10 Vacuum impedance 𝑍0 = √𝜇0 /𝜀0 376.730 313 668(57) Ω 1.5 ⋅ 10−10 Loschmidt’s number 𝑁L 2.686 780 111... ⋅ 1025 1/m3 0 at 273.15 K and 101 325 Pa Faraday’s constant 𝐹 = 𝑁A 𝑒 96 485.332 12... C/mol 0 Universal gas constant 𝑅 = 𝑁A 𝑘 8.314 462 618... J/(mol K) 0 Molar volume of an ideal gas 𝑉 = 𝑅𝑇/𝑝 22.413 969 54... l/mol 0 at 273.15 K and 101 325 Pa Rydberg constant 𝑎 𝑅∞ = 𝑚e 𝑐𝛼2 /2ℎ 10 973 731.568 160(21) m−1 1.9 ⋅ 10−12 Conductance quantum 𝐺0 = 2𝑒2 /ℎ 77.480 917 29... μS 0 Magnetic flux quantum 𝜑0 = ℎ/2𝑒 2.067 833 848... fWb 0 Josephson frequency ratio 2𝑒/ℎ 483.597 8484... THz/V 0 Von Klitzing constant ℎ/𝑒2 = 𝜇0 𝑐/2𝛼 25 812.807 45... Ω 0 Bohr magneton 𝜇B = 𝑒ℏ/2𝑚e 9.274 010 0783(28) yJ/T 3.0 ⋅ 10−10 Classical electron radius 𝑟e = 𝑒2 /4π𝜀0 𝑚e 𝑐2 2.817 940 3262(13) f m 4.5 ⋅ 10−10 Compton wavelength 𝜆 C = ℎ/𝑚e 𝑐 2.426 310 238 67(73) pm 3.0 ⋅ 10−10 of the electron 𝜆c = ℏ/𝑚e 𝑐 = 𝑟e /𝛼 0.386 159 267 96(12) pm 3.0 ⋅ 10−10 a units, measurements and constants 217 TA B L E 12 (Continued) Derived physical constants. Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. Bohr radius 𝑎 𝑎∞ = 𝑟e /𝛼2 52.917 721 0903(80) pm 1.5 ⋅ 10−10 Quantum of circulation ℎ/2𝑚e 3.636 947 5516(11) cm2 /s 3.0 ⋅ 10−10 Specific positron charge 𝑒/𝑚e 175.882 001 076(55) GC/kg 3.0 ⋅ 10−10 Cyclotron frequency 𝑓c /𝐵 = 𝑒/2π𝑚e 27.992 489 872(9) GHz/T 3.0 ⋅ 10−10 of the electron Electron magnetic moment 𝜇e −9.284 764 7043(28) yJ/T 3.0 ⋅ 10−10 𝜇e /𝜇B −1.001 159 652 181 28(18) 1.7 ⋅ 10−13 𝜇e /𝜇N −1 838.281 971 88(11) ⋅ 103 6.0 ⋅ 10−11 Electron g-factor 𝑔e −2.002 319 304 362 56(35) 1.7 ⋅ 10−13 Muon–electron mass ratio 𝑚μ /𝑚e 206.768 2830(46) 2.2 ⋅ 10−8 Muon magnetic moment 𝜇μ −4.490 448 30(10) ⋅ 10−26 J/T 2.2 ⋅ 10−8 Motion Mountain – The Adventure of Physics Muon g-factor 𝑔μ −2.002 331 8418(13) 6.3 ⋅ 10−10 Atomic mass unit 1 u = 𝑚12C /12 1.660 539 066 60(50) ⋅ 10 kg 3.0 ⋅ 10−10 −27 Proton mass 𝑚p 1.672 621 923 69(51) ⋅ 10−27 kg 3.1 ⋅ 10−10 1.007 276 466 621(53) u 5.3 ⋅ 10−11 938.272 088 16(29) MeV 3.1 ⋅ 10−10 Proton–electron mass ratio 𝑚p /𝑚e 1 836.152 673 43(11) 6.0 ⋅ 10−11 Specific proton charge 𝑒/𝑚p 9.578 833 1560(29) ⋅ 107 C/kg 3.1 ⋅ 10−10 Proton Compton wavelength 𝜆 C,p = ℎ/𝑚p 𝑐 1.321 409 855 39(40) f m 3.1 ⋅ 10−10 Nuclear magneton 𝜇N = 𝑒ℏ/2𝑚p 5.050 783 7461(15) ⋅ 10 J/T 3.1 ⋅ 10−10 −27 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Proton magnetic moment 𝜇p 1.410 606 797 36(60) ⋅ 10−26 J/T 4.2 ⋅ 10−10 𝜇p /𝜇B 1.521 032 202 30(46) ⋅ 10−3 3.0 ⋅ 10−10 𝜇p /𝜇N 2.792 847 344 63(82) 2.9 ⋅ 10−10 Proton gyromagnetic ratio 𝛾p = 2𝜇𝑝 /ℎ 42.577 478 518(18) MHz/T 4.2 ⋅ 10−10 Proton g factor 𝑔p 5.585 694 6893(16) 2.9 ⋅ 10−10 Neutron mass 𝑚n 1.674 927 498 04(95) ⋅ 10−27 kg 5.7 ⋅ 10−10 1.008 664 915 95(43) u 4.8 ⋅ 10−10 939.565 420 52(54) MeV 5.7 ⋅ 10−10 Neutron–electron mass ratio 𝑚n /𝑚e 1 838.683 661 73(89) 4.8 ⋅ 10−10 Neutron–proton mass ratio 𝑚n /𝑚p 1.001 378 419 31(49) 4.9 ⋅ 10−10 Neutron Compton wavelength 𝜆 C,n = ℎ/𝑚n 𝑐 1.319 590 905 81(75) f m 5.7 ⋅ 10−10 Neutron magnetic moment 𝜇n −0.966 236 51(23) ⋅ 10−26 J/T 2.4 ⋅ 10−7 𝜇n /𝜇B −1.041 875 63(25) ⋅ 10−3 2.4 ⋅ 10−7 𝜇n /𝜇N −1.913 042 73(45) 2.4 ⋅ 10−7 Stefan–Boltzmann constant 𝜎 = π2 𝑘4 /60ℏ3 𝑐2 56.703 744 19... nW/m K 2 4 0 Wien’s displacement constant 𝑏 = 𝜆 max 𝑇 2.897 771 955... mmK 0 58.789 257 57... GHz/K 0 Electron volt eV 0.160 217 6634... aJ 0 Bits to entropy conversion const. 𝑘 ln 2 1023 bit = 0.956 994... J/K 0 218 a units, measurements and constants TA B L E 12 (Continued) Derived physical constants. Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. TNT energy content 3.7 to 4.0 MJ/kg 4 ⋅ 10−2 𝑎. For infinite mass of the nucleus. Some useful properties of our local environment are given in the following table. TA B L E 13 Astronomical constants. Q ua nt it y Symbol Va l u e Tropical year 1900 𝑎 𝑎 31 556 925.974 7 s Tropical year 1994 𝑎 31 556 925.2 s Mean sidereal day 𝑑 23ℎ 56 4.090 53 Motion Mountain – The Adventure of Physics Average distance Earth–Sun 𝑏 149 597 870.691(30) km Astronomical unit 𝑏 AU 149 597 870 691 m Light year, based on Julian year 𝑏 al 9.460 730 472 5808 Pm Parsec pc 30.856 775 806 Pm = 3.261 634 al Earth’s mass 𝑀♁ 5.973(1) ⋅ 1024 kg Geocentric gravitational constant 𝐺𝑀 3.986 004 418(8) ⋅ 1014 m3 /s2 2 Earth’s gravitational length 𝑙♁ = 2𝐺𝑀/𝑐 8.870 056 078(16) mm Earth’s equatorial radius 𝑐 𝑅♁eq 6378.1366(1) km Earth’s polar radius 𝑐 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 𝑅♁p 6356.752(1) km Equator–pole distance 𝑐 10 001.966 km (average) Earth’s flattening 𝑐 𝑒♁ 1/298.25642(1) Earth’s av. density 𝜌♁ 5.5 Mg/m3 Earth’s age 𝑇♁ 4.50(4) Ga = 142(2) Ps Earth’s normal gravity 𝑔 9.806 65 m/s2 Earth’s standard atmospher. pressure 𝑝0 101 325 Pa Moon’s radius 𝑅v 1738 km in direction of Earth Moon’s radius 𝑅h 1737.4 km in other two directions Moon’s mass 𝑀 7.35 ⋅ 1022 kg Moon’s mean distance 𝑑 𝑑 384 401 km Moon’s distance at perigee 𝑑 typically 363 Mm, historical minimum 359 861 km Moon’s distance at apogee 𝑑 typically 404 Mm, historical maximum 406 720 km Moon’s angular size 𝑒 average 0.5181° = 31.08 , minimum 0.49°, maximum 0.55° Moon’s average density 𝜌 3.3 Mg/m3 Moon’s surface gravity 𝑔 1.62 m/s2 Moon’s atmospheric pressure 𝑝 from 10−10 Pa (night) to 10−7 Pa (day) Jupiter’s mass 𝑀 1.90 ⋅ 1027 kg a units, measurements and constants 219 TA B L E 13 (Continued) Astronomical constants. Q ua nt it y Symbol Va l u e Jupiter’s radius, equatorial 𝑅 71.398 Mm Jupiter’s radius, polar 𝑅 67.1(1) Mm Jupiter’s average distance from Sun 𝐷 778 412 020 km Jupiter’s surface gravity 𝑔 24.9 m/s2 Jupiter’s atmospheric pressure 𝑝 from 20 kPa to 200 kPa Sun’s mass 𝑀⊙ 1.988 43(3) ⋅ 1030 kg Sun’s gravitational length 2𝐺𝑀⊙ /𝑐2 2.953 250 08(5) km Heliocentric gravitational constant 𝐺𝑀⊙ 132.712 440 018(8) ⋅ 1018 m3 /s2 Sun’s luminosity 𝐿⊙ 384.6 YW Solar equatorial radius 𝑅⊙ 695.98(7) Mm Sun’s angular size 0.53∘ average; minimum on fourth of July (aphelion) 1888 , maximum on fourth of Motion Mountain – The Adventure of Physics January (perihelion) 1952 Sun’s average density 𝜌⊙ 1.4 Mg/m3 Sun’s average distance AU 149 597 870.691(30) km Sun’s age 𝑇⊙ 4.6 Ga Solar velocity 𝑣⊙g 220(20) km/s around centre of galaxy Solar velocity 𝑣⊙b 370.6(5) km/s against cosmic background copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Sun’s surface gravity 𝑔⊙ 274 m/s2 Sun’s lower photospheric pressure 𝑝⊙ 15 kPa Distance to Milky Way’s centre 8.0(5) kpc = 26.1(1.6) kal Milky Way’s age 13.6 Ga Milky Way’s size c. 1021 m or 100 kal Milky Way’s mass 1012 solar masses, c. 2 ⋅ 1042 kg Most distant galaxy cluster known SXDF-XCLJ 9.6 ⋅ 109 al 0218-0510 𝑎. Defining constant, from vernal equinox to vernal equinox; it was once used to define the second. (Remember: π seconds is about a nanocentury.) The value for 1990 is about 0.7 s less, Challenge 174 s corresponding to a slowdown of roughly 0.2 ms/a. (Watch out: why?) There is even an empirical Ref. 168 formula for the change of the length of the year over time. 𝑏. The truly amazing precision in the average distance Earth–Sun of only 30 m results from time averages of signals sent from Viking orbiters and Mars landers taken over a period of over twenty years. Note that the International Astronomical Union distinguishes the average distance Earth– Sun from the astronomical unit itself; the latter is defined as a fixed and exact length. Also the light year is a unit defined as an exact number by the IAU. For more details, see www.iau.org/ public/measuring. 𝑐. The shape of the Earth is described most precisely with the World Geodetic System. The last edition dates from 1984. For an extensive presentation of its background and its details, see the 220 a units, measurements and constants www.wgs84.com website. The International Geodesic Union refined the data in 2000. The radii and the flattening given here are those for the ‘mean tide system’. They differ from those of the ‘zero tide system’ and other systems by about 0.7 m. The details constitute a science in itself. 𝑑. Measured centre to centre. To find the precise position of the Moon at a given date, see the www.fourmilab.ch/earthview/moon_ap_per.html page. For the planets, see the page www. fourmilab.ch/solar/solar.html and the other pages on the same site. 𝑒. Angles are defined as follows: 1 degree = 1∘ = π/180 rad, 1 (first) minute = 1 = 1°/60, 1 second (minute) = 1 = 1 /60. The ancient units ‘third minute’ and ‘fourth minute’, each 1/60th of the preceding, are not in use any more. (‘Minute’ originally means ‘very small’, as it still does in modern English.) Some properties of nature at large are listed in the following table. (If you want a chal- Challenge 175 s lenge, can you determine whether any property of the universe itself is listed?) TA B L E 14 Cosmological constants. Q ua nt it y Symbol Va l u e Motion Mountain – The Adventure of Physics Cosmological constant Λ c. 1 ⋅ 10−52 m−2 𝑎 Age of the universe 𝑡0 4.333(53) ⋅ 1017 s = 13.8(0.1) ⋅ 109 a (determined from space-time, via expansion, using general relativity) Age of the universe 𝑎 𝑡0 over 3.5(4) ⋅ 1017 s = 11.5(1.5) ⋅ 109 a (determined from matter, via galaxies and stars, using quantum theory) Hubble parameter 𝑎 𝐻0 2.3(2) ⋅ 10−18 s−1 = 0.73(4) ⋅ 10−10 a−1 = ℎ0 ⋅ 100 km/s Mpc = ℎ0 ⋅ 1.0227 ⋅ 10−10 a−1 𝑎 Reduced Hubble parameter ℎ0 0.71(4) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 𝑎 2 Deceleration parameter ̈ 0 /𝐻0 −0.66(10) 𝑞0 = −(𝑎/𝑎) Universe’s horizon distance 𝑎 𝑑0 = 3𝑐𝑡0 40.0(6) ⋅ 1026 m = 13.0(2) Gpc Universe’s topology trivial up to 1026 m Number of space dimensions 3, for distances up to 1026 m Critical density 𝜌c = 3𝐻02 /8π𝐺 ℎ20 ⋅ 1.878 82(24) ⋅ 10−26 kg/m3 of the universe = 0.95(12) ⋅ 10−26 kg/m3 (Total) density parameter 𝑎 Ω0 = 𝜌0 /𝜌c 1.02(2) Baryon density parameter 𝑎 ΩB0 = 𝜌B0 /𝜌c 0.044(4) 𝑎 Cold dark matter density parameter ΩCDM0 = 𝜌CDM0 /𝜌c 0.23(4) Neutrino density parameter 𝑎 Ω𝜈0 = 𝜌𝜈0 /𝜌c 0.001 to 0.05 𝑎 Dark energy density parameter ΩX0 = 𝜌X0 /𝜌c 0.73(4) Dark energy state parameter 𝑤 = 𝑝X /𝜌X −1.0(2) Baryon mass 𝑚b 1.67 ⋅ 10−27 kg Baryon number density 0.25(1) /m3 Luminous matter density 3.8(2) ⋅ 10−28 kg/m3 Stars in the universe 𝑛s 1022±1 Baryons in the universe 𝑛b 1081±1 Microwave background temperature 𝑏 𝑇0 2.725(1) K Photons in the universe 𝑛𝛾 1089 Photon energy density 𝜌𝛾 = π2 𝑘4 /15𝑇04 4.6 ⋅ 10−31 kg/m3 a units, measurements and constants 221 TA B L E 14 (Continued) Cosmological constants. Q ua nt it y Symbol Va l u e Photon number density 410.89 /cm3 or 400 /cm3 (𝑇0 /2.7 K)3 Density perturbation amplitude √𝑆 5.6(1.5) ⋅ 10−6 Gravity wave amplitude √𝑇 < 0.71√𝑆 Mass fluctuations on 8 Mpc 𝜎8 0.84(4) Scalar index 𝑛 0.93(3) Running of scalar index d𝑛/d ln 𝑘 −0.03(2) Planck length 𝑙Pl = √ℏ𝐺/𝑐3 1.62 ⋅ 10−35 m Planck time 𝑡Pl = √ℏ𝐺/𝑐5 5.39 ⋅ 10−44 s Planck mass 𝑚Pl = √ℏ𝑐/𝐺 21.8 μg 𝑎 Instants in history 𝑡0 /𝑡Pl 8.7(2.8) ⋅ 1060 Motion Mountain – The Adventure of Physics Space-time points 𝑁0 = (𝑅0 /𝑙Pl )3 ⋅ 10244±1 inside the horizon 𝑎 (𝑡0 /𝑡Pl ) Mass inside horizon 𝑀 1054±1 kg 𝑎. The index 0 indicates present-day values. 𝑏. The radiation originated when the universe was 380 000 years old and had a temperature of about 3000 K; the fluctuations Δ𝑇0 which led to galaxy formation are today about 16 ± 4 μK = Vol. II, page 231 6(2) ⋅ 10−6 𝑇0 . copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Useful numbers π 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 375105 e 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 699959 γ 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 939923 Ref. 169 ln 2 0.69314 71805 59945 30941 72321 21458 17656 80755 00134 360255 ln 10 2.30258 50929 94045 68401 79914 54684 36420 76011 01488 628772 √10 3.16227 76601 68379 33199 88935 44432 71853 37195 55139 325216 If the number π is normal, i.e., if all digits and digit combinations in its decimal expansion appear with the same limiting frequency, then every text ever written or yet to be written, as well as every word ever spoken or yet to be spoken, can be found coded in its sequence. The property of normality has not yet been proven, although it is suspected to hold. Does this mean that all wisdom is encoded in the simple circle? No. The property is nothing special: it also applies to the number 0.123456789101112131415161718192021... Challenge 176 s and many others. Can you specify a few examples? By the way, in the graph of the exponential function e𝑥 , the point (0, 1) is the only point with two rational coordinates. If you imagine painting in blue all points on the plane with two rational coordinates, the plane would look quite bluish. Nevertheless, the graph goes through only one of these points and manages to avoid all the others. Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net a units, measurements and constants 222 Appendix B N UM BE R S A N D V E C TOR SPAC E S “ A mathematician is a machine that transforms ” coffee into theorems. Paul Erdős (b. 1913 Budapest, d. 1996 Warsaw) M athematical concepts can all be expressed in terms of ‘sets’ and ‘relations.’ Motion Mountain – The Adventure of Physics any fundamental concepts were presented in the last chapter. Why does athematics, given this simple basis, grow into a passion for certain people? How Vol. III, page 285 can sets and relations become the center of a person’s life? The mathematical appendices Ref. 170 present a few more advanced concepts as simply and vividly as possible, for all those who want to understand and to smell the passion for mathematics. Unfortunately, the passion for mathematics is not easy to spot, because like many other professions, also mathematicians hide their passions. In mathematics, this is done through formalism and apparent detachment from intuition. Good mathematical teach- ing however, puts intuition at the beginning. In this appendix we shall introduce the copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net simplest algebraic structures. The appendix in the next volume will present some more involved algebraic structures and then the most important topological structures; the third basic type of mathematical structures, order structures, are not so important in physics – with one exception: the definition of the real numbers contains an order struc- ture. Mathematicians are concerned not only with the exploration of concepts, but also with their classification. Whenever a new mathematical concept is introduced, mathem- aticians try to classify all the possible cases and types. This has been achieved most spec- tacularly for the different types of numbers, for finite simple groups and for many types of spaces and manifolds. Numbers as mathematical structures A person who can solve 𝑥2 − 92𝑦2 = 1 in less Challenge 177 ny “ than a year is a mathematician. Brahmagupta (b. 598 Sindh, d. 668) (implied: solve in integers) ” Children know: numbers are entities that can be added and multiplied. Mathematicians are more discerning. Any mathematical system with the same basic properties as the nat- ural numbers is called a semi-ring. Any mathematical system with the same basic prop- erties as the integers is called a ring. (The terms are due to David Hilbert. Both structures can also be finite rather than infinite.) More precisely, a ring (𝑅, +, ⋅) is a set 𝑅 of ele- 224 b numbers and vector spaces ments with two binary operations, called addition and multiplication, usually written + and ⋅ (the latter may simply be understood, thus without explicit notation), for which the following properties hold for all elements 𝑎, 𝑏, 𝑐 ∈ 𝑅: — 𝑅 is a commutative group with respect to addition, i.e. 𝑎 + 𝑏 ∈ 𝑅, 𝑎 + 𝑏 = 𝑏 + 𝑎, 𝑎 + 0 = 𝑎, 𝑎 + (−𝑎) = 𝑎 − 𝑎 = 0 and 𝑎 + (𝑏 + 𝑐) = (𝑎 + 𝑏) + 𝑐; — 𝑅 is closed under multiplication, i.e., 𝑎𝑏 ∈ 𝑅; — multiplication is associative, i.e., 𝑎(𝑏𝑐) = (𝑎𝑏)𝑐; — distributivity holds, i.e., 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 and (𝑏 + 𝑐)𝑎 = 𝑏𝑎 + 𝑐𝑎. Many authors add the axiom — a multiplicative unit exists, i.e., 1𝑎 = 𝑎1 = 𝑎. Defining properties such as these are called axioms. We stress that axioms are not basic beliefs, as is often stated or implied; axioms are the basic properties used in the definition of a concept: in this case, of a ring. With the last axiom, one also speaks of a unital ring. Motion Mountain – The Adventure of Physics A semi-ring is a set satisfying all the axioms of a ring, except that the existence of neutral and negative elements for addition is replaced by the weaker requirement that if 𝑎 + 𝑐 = 𝑏 + 𝑐 then 𝑎 = 𝑏. Sloppily, a semi-ring is a ring ‘without’ negative elements. To incorporate division and define the rational numbers, we need another concept. A number field or field K is a ring with — a multiplicative identity 1, such that all elements 𝑎 obey 1𝑎 = 𝑎; — at least one element different from zero; and most importantly — a (multiplicative) inverse 𝑎−1 for every element 𝑎 ≠ 0. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net A ring or field is said to be commutative if the multiplication is commutative. A non- commutative field is also called a skew field. Fields can be finite or infinite. (A field or a ring is characterized by its characteristic 𝑝. This is the smallest number of times one has to add 1 to itself to give zero. If there is no such number the characteristic is set to 0. 𝑝 is always a prime number or zero.) All finite fields are commutative. In a field, all equations of the type 𝑐𝑥 = 𝑏 and 𝑥𝑐 = 𝑏 (𝑐 ≠ 0) have solutions for 𝑥; there is a unique solution if 𝑏 ≠ 0. To sum up sloppily by focusing on the most important property, a field is a set of elements for which, together with addition, subtraction and multiplication, a division (by non-zero elements) is also defined. The rational numbers are the simplest field that incorporates the integers. The system of the real numbers is the minimal extension of the rationals which is Challenge 178 e complete and totally ordered.* Can you show that √2 is a real, but not a rational number? * A set is mathematically complete if physicists call it continuous. More precisely, a set of numbers is complete if every non-empty subset that is bounded above has a least upper bound. A set is totally ordered if there exists a binary relation ⩽ between pairs of elements such that for all elements 𝑎 and 𝑏 — if 𝑎 ⩽ 𝑏 and 𝑏 ⩽ 𝑐, then 𝑎 ⩽ 𝑐; — if 𝑎 ⩽ 𝑏 and 𝑏 ⩽ 𝑎, then 𝑎 = 𝑏; — 𝑎 ⩽ 𝑏 or 𝑏 ⩽ 𝑎 holds. In summary, a set is totally ordered if there is a binary relation that allows saying about any two elements which one is the predecessor of the other in a consistent way. This is the fundamental – and also the only – order structure used in physics. b numbers and vector spaces 225 imaginary axis 𝑧 = 𝑎 + 𝑖𝑏 = 𝑟e𝑖𝜑 𝑏 𝑟 = |𝑧| 𝜑 real axis −𝜑 𝑎 𝑟 = |𝑧| 𝑧∗ = 𝑎 − 𝑖𝑏 = 𝑟e−𝑖𝜑 F I G U R E 86 Complex numbers are points in the two-dimensional plane; a complex number 𝑧 and its conjugate 𝑧∗ can be described in cartesian form or Motion Mountain – The Adventure of Physics in polar form. In classical physics and quantum theory, it is always stressed that measurement results are and must be real numbers. But are all real numbers possible measurement results? In Challenge 179 s other words, are all measurement results just a subset of the reals? However, the concept of ‘number’ is not limited to these examples. It can be gen- Ref. 171 eralized in several ways. The simplest generalization is achieved by extending the real numbers to manifolds of more than one dimension. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net C omplex numbers In nature, complex numbers are a useful way to describe in compact form systems and situations that contain a phase. Complex numbers are thus useful to describe waves of any kind. Complex numbers form a two-dimensional manifold. A complex number is defined, in its cartesian form, by 𝑧 = 𝑎 + 𝑖𝑏, where 𝑎 and 𝑏 are real numbers, and 𝑖 is a new sym- bol, the so-called imaginary unit. Under multiplication, the generators of the complex numbers, 1 and 𝑖, obey ⋅ 1 𝑖 1 1 𝑖 (120) 𝑖 𝑖 −1 often summarized as 𝑖 = +√−1 . In a complex number 𝑧 = 𝑎 + 𝑖𝑏, 𝑎 is called the real part, and 𝑏 the complex part. They are illustrated in Figure 86. The complex conjugate 𝑧∗ , also written 𝑧,̄ of a complex number 𝑧 = 𝑎 + 𝑖𝑏 is defined as 𝑧∗ = 𝑎 − 𝑖𝑏. The absolute value |𝑧| of a complex number is defined as |𝑧| = √𝑧𝑧∗ = √𝑧∗ 𝑧 = √𝑎2 + 𝑏2 . It defines a norm on the vector space of the complex numbers. From |𝑤𝑧| = |𝑤| |𝑧| follows the two-squares theorem (𝑎12 + 𝑎22 )(𝑏12 + 𝑏22 ) = (𝑎1 𝑏1 − 𝑎2 𝑏2 )2 + (𝑎1 𝑏2 + 𝑎2 𝑏1 )2 (121) 226 b numbers and vector spaces 𝑖𝑐 𝑖ℎ = − 𝑖𝑎𝑏 𝑐 𝑎 0 𝑏 F I G U R E 87 A property of triangles easily provable with complex numbers. Motion Mountain – The Adventure of Physics valid for all real numbers 𝑎𝑖 , 𝑏𝑖 . It was already known, in its version for integers, to Dio- phantus of Alexandria in the third century CE. Complex numbers can also be written as ordered pairs (𝑎, 𝐴) of real numbers, with their addition defined as (𝑎, 𝐴) + (𝑏, 𝐵) = (𝑎 + 𝑏, 𝐴 + 𝐵) and their multiplication defined as (𝑎, 𝐴) ⋅ (𝑏, 𝐵) = (𝑎𝑏 − 𝐴𝐵, 𝑎𝐵 + 𝑏𝐴). This notation allows us to identify the complex numbers with the points on a plane or, if we prefer, to arrows in a plane. Translating the definition of multiplication into geometrical language allows us to rapidly prove certain geometrical theorems, such as the one of Figure 87. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Challenge 180 e Complex numbers 𝑎 + 𝑖𝑏 can also be represented as 2 × 2 matrices 𝑎 𝑏 ( ) with 𝑎, 𝑏 ∈ ℝ . (122) −𝑏 𝑎 Matrix addition and multiplication then correspond to complex addition and multiplic- ation. In this way, complex numbers can be represented by a special type of real matrix. Challenge 181 s What is |𝑧| in matrix language? The set ℂ of complex numbers with addition and multiplication as defined above Page 235 forms both a commutative two-dimensional field and a vector space over ℝ. In the field of complex numbers, quadratic equations 𝑎𝑧2 + 𝑏𝑧 + 𝑐 = 0 for an unknown 𝑧 always have Challenge 182 e two solutions (for 𝑎 ≠ 0 and counting multiplicity). Complex numbers can be used to describe the points of a plane. A rotation around the origin can be described by multiplication by a complex number of unit length. Other two-dimensional quantities can also be described with complex numbers. Electrical en- gineers use complex numbers to describe quantities with phases, such as alternating cur- rents or electrical fields in space. Writing complex numbers of unit length as cos 𝜃 + 𝑖 sin 𝜃 is a useful method for re- Challenge 183 e membering angle addition formulae. Since one has cos 𝑛𝜃 + 𝑖 sin 𝑛𝜃 = (cos 𝜃 + 𝑖 sin 𝜃)𝑛 , one can easily deduce formulae such as cos 2𝜃 = cos2 𝜃 − sin2 𝜃 and sin 2𝜃 = 2 sin 𝜃 cos 𝜃. Challenge 184 e By the way, the unit complex numbers form the Lie group SO(2)=U(1). b numbers and vector spaces 227 Every complex number can be written as 𝑧 = 𝑟e𝑖𝜑 . (123) This polar form of writing complex numbers is the reason for introducing them in the first place. The angle 𝜑 is called the phase; the real number 𝑟 = |𝑧| is called the absolute value or the modulus or the magnitude. When used to describe oscillations or waves, it makes sense to call 𝑟 the amplitude. The complex exponential function is periodic in 2π𝑖; in other words, we have e1 = e1+2π𝑖 , (124) which shows the property we expect from a phase angle. If one uses the last equation twice, one may write 2 2 e1 = e1+2π𝑖 = (e1+2π𝑖 )1+2π𝑖 = e(1+2π𝑖)(1+2π𝑖) = e1−4π +4π𝑖 = e1−4π . (125) Motion Mountain – The Adventure of Physics Challenge 185 e Oops, that would imply π = 0! What is wrong here? Complex numbers can also be used to describe Euclidean plane geometry. Rotations, translations and other isometries, but also reflections, glide reflections and scaling are easily described by simple operations on the complex numbers that describe the coordin- ate of points. By the way, there are exactly as many complex numbers as there are real numbers. Challenge 186 s Can you show this? copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net “ ” Love is complex: it has real and imaginary parts. Anonymous Q uaternions The positions of the points on a line can be described by real numbers. Complex num- bers can be used to describe the positions of the points of a plane. It is natural to try to generalize the idea of a number to higher-dimensional spaces. However, it turns out that no useful number system can be defined for three-dimensional space. A new num- ber system, the quaternions, can be constructed which corresponds the points of four- dimensional space, but only if the commutativity of multiplication is sacrificed. No useful number system can be defined for dimensions other than 1, 2 and 4. The quaternions were discovered by several mathematicians in the nineteenth cen- tury, among them Hamilton,* who studied them for much of his life. In fact, Max- well’s theory of electrodynamics was formulated in terms of quaternions before three- Ref. 173 dimensional vectors were used. Vol. V, page 358 Under multiplication, the quaternions ℍ form a 4-dimensional algebra over the reals * William Rowan Hamilton (b. 1805 Dublin, d. 1865 Dunsink), child prodigy and famous mathematician, named the quaternions after an expression from the Vulgate (Acts. 12: 4). 228 b numbers and vector spaces with a basis 1, 𝑖, 𝑗, 𝑘 satisfying ⋅ 1 𝑖 𝑗 𝑘 1 1 𝑖 𝑗 𝑘 𝑖 𝑖 −1 𝑘 −𝑗 (126) 𝑗 𝑗 −𝑘 −1 𝑖 𝑘 𝑘 𝑗 −𝑖 −1 These relations are also often written 𝑖2 = 𝑗2 = 𝑘2 = −1, 𝑖𝑗 = −𝑗𝑖 = 𝑘, 𝑗𝑘 = −𝑘𝑗 = 𝑖, 𝑘𝑖 = −𝑖𝑘 = 𝑗. The quaternions 1, 𝑖, 𝑗, 𝑘 are also called basic units or generators. The lack of symmetry across the diagonal of the table shows the non-commutativity of quaternionic multiplication. With the quaternions, the idea of a non-commutative product appeared for the first time in mathematics. However, the multiplication of quaternions is asso- ciative. As a consequence of non-commutativity, polynomial equations in quaternions have many more solutions than in complex numbers: just search for all solutions of the Motion Mountain – The Adventure of Physics Challenge 187 s equation 𝑋2 + 1 = 0 to convince yourself of it. Every quaternion 𝑋 can be written in the form 𝑋 = 𝑥0 + 𝑥1 𝑖 + 𝑥2 𝑗 + 𝑥3 𝑘 = 𝑥0 + 𝑣 = (𝑥0 , 𝑥1 , 𝑥2 , 𝑥3 ) = (𝑥0 , 𝑣) , (127) where 𝑥0 is called the scalar part and 𝑣 the vector part. The multiplication is thus defined as (𝑥, 𝑣)(𝑦, 𝑤) = (𝑥𝑦−𝑣⋅𝑤, 𝑥𝑤+𝑦𝑣+𝑣×𝑤). The multiplication of two general quaternions can be written as copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net (𝑎1 , 𝑏1 , 𝑐1 , 𝑑1 )(𝑎2 , 𝑏2 , 𝑐2 , 𝑑2 ) = (𝑎1 𝑎2 − 𝑏1 𝑏2 − 𝑐1 𝑐2 − 𝑑1 𝑑2 , 𝑎1 𝑏2 + 𝑏1 𝑎2 + 𝑐1 𝑑2 − 𝑑1 𝑐2 , 𝑎1 𝑐2 − 𝑏1 𝑑2 + 𝑐1 𝑎2 + 𝑑1 𝑏2 , 𝑎1 𝑑2 + 𝑏1 𝑐2 − 𝑐1 𝑏2 + 𝑑1 𝑎2 ) . (128) The conjugate quaternion 𝑋 is defined as 𝑋 = 𝑥0 − 𝑣, so that 𝑋𝑌 = 𝑌 𝑋. The norm |𝑋| of a quaternion 𝑋 is defined as |𝑋|2 = 𝑋𝑋 = 𝑋𝑋 = 𝑥20 + 𝑥21 + 𝑥22 + 𝑥23 = 𝑥20 + 𝑣2 . The norm is multiplicative, i.e., |𝑋𝑌| = |𝑋| |𝑌|. Unlike complex numbers, every quaternion is related to its complex conjugate by 𝑋 = − 12 (𝑋 + 𝑖𝑋𝑖 + 𝑗𝑋𝑗 + 𝑘𝑋𝑘) . (129) No relation of this type exists for complex numbers. In the language of physics, a complex number and its conjugate are independent variables; for quaternions, this is not the case. As a result, functions of quaternions are less useful in physics than functions of complex variables. The relation |𝑋𝑌| = |𝑋| |𝑌| implies the four-squares theorem (𝑎12 + 𝑎22 + 𝑎32 + 𝑎42 )(𝑏12 + 𝑏22 + 𝑏32 + 𝑏42 ) = (𝑎1 𝑏1 − 𝑎2 𝑏2 − 𝑎3 𝑏3 − 𝑎4 𝑏4 )2 + (𝑎1 𝑏2 + 𝑎2 𝑏1 + 𝑎3 𝑏4 − 𝑎4 𝑏3 )2 + (𝑎1 𝑏3 + 𝑎3 𝑏1 + 𝑎4 𝑏2 − 𝑎2 𝑏4 )2 + (𝑎1 𝑏4 + 𝑎4 𝑏1 + 𝑎2 𝑏3 − 𝑎3 𝑏2 )2 (130) b numbers and vector spaces 229 𝛼/2 𝑙 π − 𝛾/2 𝑛 𝛽/2 𝑚 F I G U R E 88 Combinations of rotations. Motion Mountain – The Adventure of Physics valid for all real numbers 𝑎𝑖 and 𝑏𝑖 , and thus also for any set of eight integers. It was discovered in 1748 by Leonhard Euler (1707–1783) when trying to prove that each integer is the sum of four squares. (The latter fact was proved only in 1770, by Joseph Lagrange.) Hamilton thought that a quaternion with zero scalar part, which he simply called a vector (a term which he invented), could be identified with an ordinary three- dimensional translation vector; but this is wrong. Such a quaternion is now called a pure, or homogeneous, or imaginary quaternion. The product of two pure quaternions copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 𝑉 = (0, 𝑣) and 𝑊 = (0, 𝑤) is given by 𝑉𝑊 = (−𝑣 ⋅ 𝑤, 𝑣 × 𝑤), where ⋅ denotes the scalar product and × denotes the vector product. Note that any quaternion can be written as the ratio of two pure quaternions. In reality, a pure quaternion (0, 𝑣) does not behave like a translation vector under coordinate transformations; in fact, a pure quaternion represents a rotation by the angle Challenge 188 ny π or 180° around the axis defined by the direction 𝑣 = (𝑣𝑥, 𝑣𝑦 , 𝑣𝑧 ). It turns out that in three-dimensional space, a general rotation about the origin can be described by a unit quaternion 𝑄, also called a normed quaternion, for which |𝑄| = 1. Such a quaternion can be written as (cos 𝜃/2, 𝑛 sin 𝜃/2), where 𝑛 = (𝑛𝑥 , 𝑛𝑦 , 𝑛𝑧 ) is the normed vector describing the direction of the rotation axis and 𝜃 is the rotation angle. Such a unit quaternion 𝑄 = (cos 𝜃/2, 𝑛 sin 𝜃/2) rotates a pure quaternion 𝑉 = (0, 𝑣) into another pure quaternion 𝑊 = (0, 𝑤) given by 𝑊 = 𝑄𝑉𝑄∗ . (131) Thus, if we use pure quaternions such as 𝑉 or 𝑊 to describe positions, we can use unit quaternions to describe rotations and to calculate coordinate changes. The concatenation of two rotations is then given by the product of the corresponding unit quaternions. Indeed, a rotation by an angle 𝛼 about the axis 𝑙 followed by a rotation by an angle 𝛽 about the axis 𝑚 gives a rotation by an angle 𝛾 about the axis 𝑛, with the values determined by (cos 𝛾/2, sin 𝛾/2𝑛) = (cos 𝛽/2, sin 𝛽/2𝑚)(cos 𝛼/2, sin 𝛼/2𝑙) . (132) 230 b numbers and vector spaces 𝑘 𝑖 𝑗 1 𝑘 𝑗 𝑖 palm back of right of right hand hand F I G U R E 89 The top and Motion Mountain – The Adventure of Physics back of the right hand, and the quaternions. One way to show the result graphically is given in Figure 88. By drawing a triangle on a unit sphere, and taking care to remember the factor 1/2 in the angles, the combination of two rotations can be simply determined. The interpretation of quaternions as rotations is also illustrated, in a somewhat differ- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Ref. 174 ent way, in the motion of any hand. To see this, take a green marker and write the letters 1, 𝑖, 𝑗 and 𝑘 on your hand as shown in Figure 89. Defining the three possible 180° rota- tion axes as shown in the figure and taking concatenation as multiplication, the motion Challenge 189 e of the right hand follows the same ‘laws’ as those of pure unit quaternions. (One needs to distinguish +𝑖 and −𝑖, and the same for the other units, by the sense of the arm twist. And the result of a multiplication is that letter that can be read by a person facing you.) Challenge 190 s You can show that 𝑖2 = 𝑗2 = 𝑘2 = −1, that 𝑖4 = 1, and conform all other quaternion rela- tions. The model also shows that the rotation angle of the arm is half the rotation angle of the corresponding quaternion. In other words, quaternions can be used to describe the belt trick, if the multiplication 𝑉𝑊 of two quaternions is taken to mean that rotation Page 130 𝑉 is performed after rotation 𝑊. Quaternions, like human hands, thus behave like a spin 1/2 particle. Quaternions and spinors are isomorphic. The reason for the half-angle behaviour of rotations can be specified more precisely using mathematical language. The rotations in three dimensions around a point form the ‘special orthogonal group’ in three dimensions, which is called SO(3). But the motions of a hand attached to a shoulder via an arm form a different group, isomorphic to the Lie Vol. V, page 369 group SU(2). The difference is due to the appearance of half angles in the parametrization of rotations; indeed, the above parametrizations imply that a rotation by 2π corresponds to a multiplication by −1. Only in the twentieth century was it realized that there ex- ist fundamental physical observables that behaves like hands attached to arms: they are Page 130 called spinors. More on spinors can be found in the section on permutation symmetry, where belts are used as an analogy as well as arms. In short, the group SU(2) formed by b numbers and vector spaces 231 Ref. 175 the unit quaternions is the double cover of the rotation group SO(3). The simple representation of rotations and positions with quaternions is used by com- puter programmes in robotics, in astronomy and in flight simulation. In the software used to create three-dimensional images and animations, visualization software, qua- ternions are often used to calculate the path taken by repeatedly reflected light rays and thus give surfaces a realistic appearance. The algebra of the quaternions is the only associative, non-commutative, finite-di- mensional normed algebra with an identity over the field of real numbers. Quaternions form a non-commutative field, i.e., a skew field, in which the inverse of a quaternion 𝑋 is 𝑋/|𝑋|. We can therefore define division of quaternions (while being careful to distin- guish 𝑋𝑌−1 and 𝑌−1 𝑋). Therefore quaternions are said to form a division algebra. In fact, the quaternions ℍ, the complex numbers ℂ and the reals ℝ are the only three finite- dimensional associative division algebras. In other words, the skew-field of quaternions is the only finite-dimensional real associative non-commutative algebra without divisors of zero. The centre of the quaternions, i.e., the set of quaternions that commute with all Motion Mountain – The Adventure of Physics other quaternions, is just the set of real numbers. Quaternions can be represented as matrices of the form 𝐴 𝐵 ( ∗ ) with 𝐴, 𝐵 ∈ ℂ thus 𝐴 = 𝑎 + 𝑖𝑏, 𝐵 = 𝑐 + 𝑖𝑑 , (133) −𝐵 𝐴∗ or, alternatively, as 𝑎 𝑏 𝑐 𝑑 −𝑏 𝑎 −𝑑 𝑐 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ( ) with 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ , (134) −𝑐 𝑑 𝑎 −𝑏 −𝑑 −𝑐 𝑏 𝑎 where the quaternion 𝑋 then is given as 𝑋 = 𝐴 + 𝐵𝑗 = 𝑎 + 𝑖𝑏 + 𝑗𝑐 + 𝑘𝑑. Matrix addition and multiplication then corresponds to quaternionic addition and multiplication. The generators of the quaternions can be realized as 1 : 𝜎0 , 𝑖 : −𝑖𝜎1 , 𝑗 : −𝑖𝜎2 , 𝑘 : −𝑖𝜎3 (135) where the 𝜎𝑛 are the Pauli spin matrices.* * The Pauli spin matrices are the complex Hermitean matrices 1 0 0 1 0 −𝑖 1 0 𝜎0 = 1 = ( ) , 𝜎1 = ( ) , 𝜎2 = ( ) , 𝜎3 = ( ) (136) 0 1 1 0 𝑖 0 0 −1 all of whose eigenvalues are ±1; they satisfy the relations [𝜎𝑖 , 𝜎𝑘 ]+ = 2 𝛿𝑖𝑘 and [𝜎𝑖 , 𝜎𝑘 ] = 2𝑖 𝜀𝑖𝑘𝑙 𝜎𝑙 . The linear combinations 𝜎± = 12 (𝜎1 ± 𝜎2 ) are also frequently used. By the way, another possible representation of the quaternions is 𝑖 : 𝑖𝜎3 , 𝑗 : 𝑖𝜎2 , 𝑘 : 𝑖𝜎1 . 232 b numbers and vector spaces Real 4 × 4 representations are not unique, as the alternative representation 𝑎 𝑏 −𝑑 −𝑐 −𝑏 𝑎 −𝑐 𝑑 ( ) (137) 𝑑 𝑐 𝑎 𝑏 𝑐 −𝑑 −𝑏 𝑎 Challenge 191 ny shows. No representation of quaternions by 3 × 3 matrices is possible. These matrices contain real and complex elements, which pose no special problems. In contrast, when matrices with quaternionic elements are constructed, care has to be taken, because quaternionic multiplication is not commutative, so that simple relations such as tr𝐴𝐵 = tr𝐵𝐴 are not generally valid. What can we learn from quaternions about the description of nature? First of all, we see that binary rotations are similar to positions, and thus to translations: all are rep- resented by 3-vectors. Are rotations the basic operations of nature? Is it possible that Motion Mountain – The Adventure of Physics translations are only ‘shadows’ of rotations? The connection between translations and Vol. VI, page 174 rotations is investigated in the last volume of our mountain ascent. When Maxwell wrote down his equations of electrodynamics, he used quaternion Vol. III, page 76 notation. (The now usual 3-vector notation was introduced later by Hertz and Heaviside.) The equations can be written in various ways using quaternions. The simplest is achieved Ref. 173 when one keeps a distinction between √−1 and the units 𝑖, 𝑗, 𝑘 of the quaternions. One Challenge 192 s then can write all of electrodynamics in a single equation: 𝑄 d𝐹 = − (138) copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 𝜀0 where 𝐹 is the generalized electromagnetic field and 𝑄 the generalized charge. These are defined by 𝐹 = 𝐸 + √−1 𝑐𝐵 𝐸 = 𝑖𝐸𝑥 + 𝑗𝐸𝑦 + 𝑘𝐸𝑧 𝐵 = 𝑖𝐵𝑥 + 𝑗𝐵𝑦 + 𝑘𝐵𝑧 (139) d = 𝛿 + √−1 ∂𝑡 /𝑐 𝛿 = 𝑖∂𝑥 + 𝑗∂𝑦 + 𝑘∂𝑧 𝑄 = 𝜌 + √−1 𝐽/𝑐 where the fields 𝐸 and 𝐵 and the charge distributions 𝜌 and 𝐽 have the usual meanings. The content of equation (138) for the electromagnetic field is exactly the same as the usual formulation. Despite their charm and their four-dimensionality, quaternions do not seem to be useful for the reformulation of special relativity; the main reason for this is the sign in the expression for their norm. Therefore, relativity and space-time are usually described using real numbers. And even if quaternions were useful, they would not provide addi- tional insights into physics or into nature. b numbers and vector spaces 233 Octonions In the same way that quaternions are constructed from complex numbers, octonions can be constructed from quaternions. They were first investigated by Arthur Cayley (1821–1895). Under multiplication, octonions (or octaves) are the elements of an eight- dimensional algebra over the reals with the generators 1, 𝑖𝑛 with 𝑛 = 1 . . . 7 satisfying ⋅ 1 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 1 1 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 𝑖7 𝑖1 𝑖1 −1 𝑖3 −𝑖2 𝑖5 −𝑖4 𝑖7 −𝑖6 𝑖2 𝑖2 −𝑖3 −1 𝑖1 −𝑖6 𝑖7 𝑖4 −𝑖5 𝑖3 𝑖3 𝑖2 −𝑖1 −1 𝑖7 𝑖6 −𝑖5 −𝑖4 (140) 𝑖4 𝑖4 −𝑖5 𝑖6 −𝑖7 −1 𝑖1 −𝑖2 𝑖3 𝑖5 𝑖5 𝑖4 −𝑖7 −𝑖6 −𝑖1 −1 𝑖3 𝑖2 𝑖6 𝑖6 −𝑖7 −𝑖4 𝑖5 𝑖2 −𝑖3 −1 𝑖1 𝑖7 𝑖7 𝑖6 𝑖5 𝑖4 −𝑖3 −𝑖2 −𝑖1 −1 Motion Mountain – The Adventure of Physics In fact, 479 other, equivalent multiplication tables are also possible. This algebra is called the Cayley algebra; it has an identity and a unique division. The algebra is non- commutative, and also non-associative. It is, however, alternative, meaning that for all elements 𝑥 and 𝑦, one has 𝑥(𝑥𝑦) = 𝑥2 𝑦 and (𝑥𝑦)𝑦 = 𝑥𝑦2 : a property somewhat weaker than associativity. It is the only 8-dimensional real alternative algebra without zero di- visors. Because it is not associative, the set 𝕆 of all octonions does not form a field, nor even a ring, so that the old designation of ‘Cayley numbers’ has been abandoned. The octonions are the most general hypercomplex ‘numbers’ whose norm is multiplicative. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Its generators obey (𝑖𝑛𝑖𝑚 )𝑖𝑙 = ±𝑖𝑛(𝑖𝑚 𝑖𝑙 ), where the minus sign, which shows the non- associativity, is valid for combinations of indices that are not quaternionic, such as 1-2-4. Octonions can be represented as matrices of the form 𝐴 𝐵 ( ) where 𝐴, 𝐵 ∈ ℍ , or as real 8 × 8 matrices. (141) −𝐵̄ 𝐴̄ Matrix multiplication then gives the same result as octonionic multiplication. 234 b numbers and vector spaces The relation |𝑤𝑧| = |𝑤| |𝑧| allows one to deduce the impressive eight-squares theorem (𝑎12 + 𝑎22 + 𝑎32 + 𝑎42 + 𝑎52 + 𝑎62 + 𝑎72 + 𝑎82 )(𝑏12 + 𝑏22 + 𝑏32 + 𝑏42 + 𝑏52 + 𝑏62 + 𝑏72 + 𝑏82 ) = (𝑎1 𝑏1 − 𝑎2 𝑏2 − 𝑎3 𝑏3 − 𝑎4 𝑏4 − 𝑎5 𝑏5 − 𝑎6 𝑏6 − 𝑎7 𝑏7 − 𝑎8 𝑏8 )2 + (𝑎1 𝑏2 + 𝑎2 𝑏1 + 𝑎3 𝑏4 − 𝑎4 𝑏3 + 𝑎5 𝑏6 − 𝑎6 𝑏5 + 𝑎7 𝑏8 − 𝑎8 𝑏7 )2 + (𝑎1 𝑏3 − 𝑎2 𝑏4 + 𝑎3 𝑏1 + 𝑎4 𝑏2 − 𝑎5 𝑏7 + 𝑎6 𝑏8 + 𝑎7 𝑏5 − 𝑎8 𝑏6 )2 + (𝑎1 𝑏4 + 𝑎2 𝑏3 − 𝑎3 𝑏2 + 𝑎4 𝑏1 + 𝑎5 𝑏8 + 𝑎6 𝑏7 − 𝑎7 𝑏6 − 𝑎8 𝑏5 )2 + (𝑎1 𝑏5 − 𝑎2 𝑏6 + 𝑎3 𝑏7 − 𝑎4 𝑏8 + 𝑎5 𝑏1 + 𝑎6 𝑏2 − 𝑎7 𝑏3 + 𝑎8 𝑏4 )2 + (𝑎1 𝑏6 + 𝑎2 𝑏5 − 𝑎3 𝑏8 − 𝑎4 𝑏7 − 𝑎5 𝑏2 + 𝑎6 𝑏1 + 𝑎7 𝑏4 + 𝑎8 𝑏3 )2 + (𝑎1 𝑏7 − 𝑎2 𝑏8 − 𝑎3 𝑏5 + 𝑎4 𝑏6 + 𝑎5 𝑏3 − 𝑎6 𝑏4 + 𝑎7 𝑏1 + 𝑎8 𝑏2 )2 + (𝑎1 𝑏8 + 𝑎2 𝑏7 + 𝑎3 𝑏6 + 𝑎4 𝑏5 − 𝑎5 𝑏4 − 𝑎6 𝑏3 − 𝑎7 𝑏2 + 𝑎8 𝑏1 )2 (142) Motion Mountain – The Adventure of Physics valid for all real numbers 𝑎𝑖 and 𝑏𝑖 and thus in particular also for all integers. (There are many variations of this expression, with different possible sign combinations.) The theorem was discovered in 1818 by Carl Ferdinand Degen (1766–1825), and then redis- covered in 1844 by John Graves and in 1845 by Arthur Cayley. There is no generalization to higher numbers of squares, a fact proved by Adolf Hurwitz (1859–1919) in 1898. The octonions can be used to show that a vector product can be defined in more than three dimensions. A vector product or cross product is an operation × satisfying 𝑢 × 𝑣 = −𝑣 × 𝑢 anticommutativity copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net (𝑢 × 𝑣) 𝑤 = 𝑢 (𝑣 × 𝑤) exchange rule. (143) Using the definition 1 𝑋 × 𝑌 = (𝑋𝑌 − 𝑌𝑋) , (144) 2 the cross products of imaginary quaternions, i.e., of quaternions of the type (0, 𝑢), are again imaginary, and correspond to the usual, three-dimensional vector product, thus Ref. 171 fulfilling (143). Interestingly, it is possible to use definition (144) for octonions as well. Challenge 193 e In that case, the product of imaginary octonions is also imaginary, and (143) is again satisfied. In fact, this is the only other non-trivial example of a vector product. In summary: A vector product exists only in three and in seven dimensions. Many schol- ars have conjectured that this relation is connected with a possible ten-dimensionality of nature; however, these speculations have not met with any success. The symmetries of the forces in nature lead to a well-known question. The unit com- plex numbers from the Lie group U(1) and the unit quaternions the Lie group SU(2). Do Challenge 194 s the unit octonions form the Lie group SU(3)? Other t ypes of numbers The process of constructing new systems of hypercomplex ‘numbers’ or real algebras by ‘doubling’ a given one can be continued ad infinitum. However, octonions, sedenions and b numbers and vector spaces 235 all the following doublings are neither rings nor fields, but only non-associative algeb- ras with unity. Other finite-dimensional algebras with unit element over the reals, once called hypercomplex ‘numbers’, can also be defined: they include the so-called ‘dual numbers’, ‘double numbers’, ‘Clifford–Lifshitz numbers’ etc. They play no role in phys- ics. Mathematicians have also defined number fields which have ‘one and a bit’ dimen- sions, such as algebraic number fields. There is also a generalization of the concept of Ref. 176 integers to the complex domain: the Gaussian integers, defined as 𝑛 + 𝑖𝑚, where 𝑛 and 𝑚 are ordinary integers. Gauss even defined what are now known as Gaussian primes. (Can Challenge 195 s you find out how?) They are not used in the description of nature, but are important in number theory, the exploration of the properties of integers. Physicists used to call quantum-mechanical operators ‘q-numbers.’ But this term has now fallen out of fashion. Another way in which the natural numbers can be extended is to include numbers Ref. 177 larger than infinite. The most important such classes of transfinite number are the ordin- Motion Mountain – The Adventure of Physics Vol. III, page 293 als, the cardinals and the surreals. The ordinals are essentially an extension of the integers beyond infinity, whereas the surreals are a continuous extension of the reals, also bey- ond infinity. Loosely speaking, among the transfinites, the ordinals have a similar role as the integers have among the reals; the surreals fill in all the gaps between the ordinals, like the reals do for integers. Interestingly, many series that diverge in ℝ converge in the Challenge 196 ny surreals. Can you find one example? The surreals include infinitely small numbers, as do the numbers of nonstandard Ref. 171 analysis, also called hyperreals. In both number systems, in contrast to real numbers, the numbers 1 and 0.999 999... (where an infinite, but hyperfinite string of nines is im- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net plied) do not coincide, but are separated by infinitely many other numbers. We explored Vol. III, page 295 surreals earlier on. Nonstandard numbers can be used to define the infinitesimals used Ref. 172 in integration and differentiation, even at secondary school level. From vector spaces to Hilbert spaces Vector spaces, also called linear spaces, are mathematical generalizations of certain as- pects of the intuitive three-dimensional space. A set of elements any two of which can be added together and any one of which can be multiplied by a number is called a vector space, if the result is again in the set and the usual rules of calculation hold. More precisely, a vector space over a number field 𝐾 is a set of elements, called vectors, for which a vector addition and a scalar multiplication is defined, such that for all vectors 𝑎, 𝑏, 𝑐 and for all numbers 𝑠 and 𝑟 from 𝐾 one has (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐) = 𝑎 + 𝑏 + 𝑐 associativity of vector addition 𝑛+𝑎 = 𝑎 existence of null vector (−𝑎) + 𝑎 = 𝑛 existence of negative vector (145) 1𝑎 = 𝑎 regularity of scalar multiplication (𝑠 + 𝑟)(𝑎 + 𝑏) = 𝑠𝑎 + 𝑠𝑏 + 𝑟𝑎 + 𝑟𝑏 complete distributivity of scalar multiplication If the field 𝐾, whose elements are called scalars in this context, is taken to be the real (or 236 b numbers and vector spaces complex, or quaternionic) numbers, one speaks of a real (or complex, or quaternionic) vector space. Vector spaces are also called linear vector spaces or simply linear spaces. The complex numbers, the set of all real functions defined on the real line, the set of all polynomials, the set of matrices with a given number of rows and columns, all form vector spaces. In mathematics, a vector is thus a more general concept than in physics. Challenge 197 s (What is the simplest possible mathematical vector space?) In physics, the term ‘vector’ is reserved for elements of a more specialized type of vector space, namely normed inner product spaces. To define these, we first need the concept of a metric space. A metric space is a set with a metric, i.e., a way to define distances between elements. A real function 𝑑(𝑎, 𝑏) between elements is called a metric if 𝑑(𝑎, 𝑏) ⩾ 0 positivity of metric 𝑑(𝑎, 𝑏) + 𝑑(𝑏, 𝑐) ⩾ 𝑑(𝑎, 𝑐) triangle inequality (146) 𝑑(𝑎, 𝑏) = 0 if and only if 𝑎 = 𝑏 regularity of metric Motion Mountain – The Adventure of Physics A non-trivial example is the following. We define a special distance 𝑑 between cities. If the two cities lie on a line going through Paris, we use the usual distance. In all other cases, we define the distance 𝑑 by the shortest distance from one to the other travelling Challenge 198 s via Paris. This strange method defines a metric between all cities in France, the so-called French railroad distance. A normed vector space is a linear space with a norm, or ‘length’, associated to each a vector. A norm is a non-negative number ‖𝑎‖ defined for each vector 𝑎 with the properties copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ‖𝑟𝑎‖ = |𝑟| ‖𝑎‖ linearity of norm ‖𝑎 + 𝑏‖ ⩽ ‖𝑎‖ + ‖𝑏‖ triangle inequality (147) ‖𝑎‖ = 0 only if 𝑎 = 0 regularity Challenge 199 ny Usually there are many ways to define a norm for a given vector space. Note that a norm can always be used to define a metric by setting 𝑑(𝑎, 𝑏) = ‖𝑎 − 𝑏‖ (148) so that all normed spaces are also metric spaces. This is the natural distance definition (in contrast to unnatural ones like that between French cities given above). The norm is often defined with the help of an inner product. Indeed, the most special class of linear spaces are the inner product spaces. These are vector spaces with an inner product, also called scalar product ⋅ (not to be confused with the scalar multiplication!) b numbers and vector spaces 237 which associates a number to each pair of vectors. An inner product space over ℝ satisfies 𝑎⋅𝑏=𝑏⋅𝑎 commutativity of scalar product (𝑟𝑎) ⋅ (𝑠𝑏) = 𝑟𝑠(𝑎 ⋅ 𝑏) bilinearity of scalar product (𝑎 + 𝑏) ⋅ 𝑐 = 𝑎 ⋅ 𝑐 + 𝑏 ⋅ 𝑐 left distributivity of scalar product 𝑎 ⋅ (𝑏 + 𝑐) = 𝑎 ⋅ 𝑏 + 𝑎 ⋅ 𝑐 right distributivity of scalar product (149) 𝑎⋅𝑎⩾0 positivity of scalar product 𝑎 ⋅ 𝑎 = 0 if and only if 𝑎 = 0 regularity of scalar product for all vectors 𝑎, 𝑏, 𝑐 and all scalars 𝑟, 𝑠. A real inner product space of finite dimension is also called a Euclidean vector space. The set of all velocities, the set of all positions, or the set of all possible momenta form such spaces. An inner product space over ℂ satisfies* Motion Mountain – The Adventure of Physics 𝑎⋅𝑏=𝑏⋅𝑎=𝑏⋅𝑎 Hermitean property (𝑟𝑎) ⋅ (𝑠𝑏) = 𝑟𝑠(𝑎 ⋅ 𝑏) sesquilinearity of scalar product (𝑎 + 𝑏) ⋅ 𝑐 = 𝑎 ⋅ 𝑐 + 𝑏 ⋅ 𝑐 left distributivity of scalar product 𝑎 ⋅ (𝑏 + 𝑐) = 𝑎 ⋅ 𝑏 + 𝑎 ⋅ 𝑐 right distributivity of scalar product (150) 𝑎⋅𝑎⩾0 positivity of scalar product 𝑎 ⋅ 𝑎 = 0 if and only if 𝑎 = 0 regularity of scalar product for all vectors 𝑎, 𝑏, 𝑐 and all scalars 𝑟, 𝑠. A complex inner product space (of finite di- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net mension) is also called a unitary or Hermitean vector space. If the inner product space is Page 224 complete, it is called, especially in the infinite-dimensional complex case, a Hilbert space. The space of all possible states of a quantum system forms a Hilbert space. All inner product spaces are also metric spaces, and thus normed spaces, if the metric is defined by 𝑑(𝑎, 𝑏) = √(𝑎 − 𝑏) ⋅ (𝑎 − 𝑏) . (151) Only in the context of an inner product spaces we can speak about angles (or phase differences) between vectors, as we are used to in physics. Of course, like in normed spaces, inner product spaces also allows us to speak about the length of vectors and to define a basis, the mathematical concept necessary to define a coordinate system. Which Challenge 200 s vector spaces or inner product spaces are of importance in physics? The dimension of a vector space is the number of linearly independent basis vectors. Challenge 201 s Can you define these terms precisely? A Hilbert space is a real or complex inner product space that is also a complete met- ric space. In other terms, in a Hilbert space, distances vary continuously and behave as naively expected. Hilbert spaces usually, but not always, have an infinite number of di- mensions. * Two inequivalent forms of the sesquilinearity axiom exist. The other is (𝑟𝑎) ⋅ (𝑠𝑏) = 𝑟𝑠(𝑎 ⋅ 𝑏). The term sesquilinear is derived from Latin and means for ‘one-and-a-half-linear’. 238 b numbers and vector spaces The definition of Hilbert spaces and vector spaces assume continuous sets to start Challenge 202 s with. If nature would not be continuous, could one still use the concepts? Mathematical curiosities and fun challenges Mathematics provides many counter-intuitive results. Reading a good book on the topic, such as Bernard R . Gelbaum & John M. H. Olmsted, Theorems and Counter- examples in Mathematics, Springer, 1993, can help you sharpen your mind and make you savour the beauty of mathematics even more. ∗∗ It is possible to draw a curve that meets all points in a square or all points in a cube. This is shown, for example, in the text Hans Sagan, Space Filling Curves, Springer Verlag, 1994. As a result, the distinction between one, two and three dimensions is blurred in pure mathematics. In physics however, dimensions are clearly and well-defined; every object in nature has three dimensions. Motion Mountain – The Adventure of Physics Challenge 203 e ∗∗ Challenge 204 ny Show that two operators 𝐴 and 𝐵 obey 1 e𝐴 e𝐵 = exp(𝐴 + 𝐵 + [𝐴, 𝐵] 2 1 1 + [[𝐴, 𝐵], 𝐵] − [[𝐴, 𝐵], 𝐴] 12 12 1 1 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net − [𝐵, [𝐴, [𝐴, 𝐵]]] − [𝐴, [𝐵, [𝐴, 𝐵]]] 48 48 + ...) (152) for most operators 𝐴 and 𝐵. This result is often called the Baker–Campbell–Hausdorff formula or the BCH formula. C HA L L E NG E H I N T S A N D S OLU T ION S “ Never make a calculation before you know the ” answer. John Wheeler’s motto Challenge 1, page 10: Do not hesitate to be demanding and strict. The next edition of the text will benefit from it. Motion Mountain – The Adventure of Physics Challenge 2, page 16: Classical physics fails in explaining any material property, such as colour or softness. Material properties result from nature’s interactions; they are inevitably quantum. Explanations of material properties require, without exception, the use of particles and their quantum properties. Challenge 3, page 17: Classical physics allows any observable to change smoothly with time. In classical physics, there is no minimum value for any observable physical quantity. Challenge 4, page 20: The higher the mass, the smaller the motion fuzziness induced by the quantum of action, because action is mass times speed times distance: For a large mass, the speed and distance variations are small. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Challenge 5, page 20: The simplest time is √𝐺ℏ/𝑐5 . The numerical factor is obviously not fixed; it is changed later on. Using 4𝐺 instead of 𝐺 the time becomes the shortest time measurable in nature. Challenge 7, page 21: The electron charge is special to the electromagnetic interactions; it does not take into account the nuclear interactions or gravity. It is unclear why the length defined with the elementary charge 𝑒 should be of importance for neutral systems or for the vacuum. On the other hand, the quantum of action ℏ is valid for all interactions and all observations. In addition, we can argue that the two options to define a fundamental length – with the quantum of action and with the quantum of charge – are not too different: the electron charge is related to the quantum of action by 𝑒 = √4π𝜀0 𝛼𝑐ℏ . The two length scales defined by the two options differ only by a factor near 11.7. In fact, both scales are quantum scales. Challenge 8, page 21: On purely dimensional grounds, the radius of an atom must be ℏ2 4π𝜀0 𝑟≈ , (153) 𝑚e 𝑒2 Page 186 which is about 53 nm. Indeed, this guess is excellent: it is just the Bohr radius. Challenge 9, page 21: Due to the quantum of action, atoms in all people, be they giants or dwarfs, have the same size. This implies that giants cannot exist, as was shown already by Galileo. Vol. I, page 338 The argument is based on the given strength of materials; and a same strength everywhere is equivalent to the same properties of atoms everywhere. That dwarfs cannot exist is due to a sim- ilar reason; nature is not able to make people smaller than usual (even in the womb they differ markedly from adults) as this would require smaller atoms. 240 challenge hints and solutions Challenge 12, page 27: A disappearance of a mass 𝑚 in a time Δ𝑡 is an action change 𝑐2 𝑚Δ𝑡. That is much larger than ℏ for all objects of everyday life. Challenge 14, page 29: Tunnelling of a lion would imply action values 𝑆 of the order of 𝑆 = 100 kgm2 /s ≫ ℏ. This cannot happen spontaneously. Challenge 15, page 30: Every memory, be it human memory or an electronic computer memory, must avoid decay. And decay can only be avoided through high walls and low tunnelling rates. Challenge 16, page 30: Yes! Many beliefs and myths – from lottery to ghosts – are due to the neglect of quantum effects. Challenge 17, page 30: Perfectly continuous flow is in contrast to the fuzziness of motion in- duced by the quantum of action. Challenge 18, page 31: The impossibility of following two particles along their path appears when their mutual distance 𝑑 is smaller than their position indeterminacy due to their relat- ive momentum 𝑝, thus when 𝑑 < ℏ/𝑝. Check the numbers with electrons, atoms, molecules, bacteria, people and galaxies. Challenge 19, page 31: Also photons are indistinguishable. See page 63. Motion Mountain – The Adventure of Physics Challenge 21, page 36: In the material that forms the escapement mechanism. Challenge 22, page 36: Growth is not proportional to light intensity or to light frequency, but shows both intensity and frequency thresholds. These are quantum effects. Challenge 23, page 36: All effects mentioned above, such as tunnelling, interference, decay, transformation, non-emptiness of the vacuum, indeterminacy and randomness, are also ob- served in the nuclear domain. Challenge 24, page 37: This is not evident from what was said so far, but it turns out to be cor- rect. In fact, there is no other option, as you will see when you try to find one. Challenge 25, page 37: Tom Thumb is supposedly as smart as a normal human. But a brain can- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net not be scaled down. Fractals contradict the existence of Planck’s length, and Moore’s law con- tradicts the existence of atoms. Challenge 26, page 37: The total angular momentum counts, including the orbital angular mo- mentum. The orbital angular momentum 𝐿 is given, using the radius and the linear momentum, 𝐿 = 𝑟 × 𝑝. The total angular momentum is a multiple of ℏ. Challenge 27, page 37: Yes, we could have! Challenge 28, page 37: That is just the indeterminacy relation. Bohr expanded this idea to all sort of other pairs of concepts, more in the philosophical domain, such as clarity and precision of explanations: both cannot be high at the same time. Challenge 29, page 39: The big bang cannot have been an event, for example. Challenge 32, page 45: Charged photons would be deflected by electric of magnetic fields; in particular, they would not cross undisturbed. This is not observed. Massive photons would be deflected by masses, such as the Sun, much more than is observed. Challenge 34, page 45: To measure momentum, we need a spatially extended measurement device; to measure position, we need a localized measurement device. Challenge 35, page 47: Photons are elementary because they realize the minimum action, be- cause they cannot decay, because they cannot be deformed or split, because they have no mass, no electric charge and no other quantum number, and because they appear in the Lagrangian of quantum electrodynamics. Challenge 36, page 50: The measured electric fields and photon distribution are shown in the famous graphs reproduced in Figure 90. challenge hints and solutions 241 Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net F I G U R E 90 Left, from top to bottom: the electric ﬁeld and its fuzziness measured for a coherent state, for a squeezed vacuum state, for a phase-squeezed state, for a mixed, quadrature-squeezed state and for an amplitude-squeezed state, all with a small number of photons. Right: the corresponding photon number distributions for the uppermost four states. (© G. Breitenbach/Macmillan, from Ref. 19) Challenge 38, page 50: This is an unclearly posed problem. The radiation is thermal, but the photon number depends on the volume under discussion. Challenge 40, page 56: Radio photons can be counted using optical pumping experiments in which atomic states are split by a small, ‘radio-wavelength’ amount, with the help of magnetic fields. Also caesium clocks detect radio photons with optical means. The Josephons effect and magnetic resonance imaging are additional detection methods for radio photons. Challenge 41, page 57: To be observable to the eye, the interference fringes need to be visible for around 0.1 s. That implies a maximum frequency difference between the two beams of around 10 Hz. This is achievable only if either a single beam is split into two or if the two beams come from high-precision, stabilized lasers. Challenge 42, page 62: Implicit in the arrow model is the idea that one quantum particle is de- scribed by one arrow. 242 challenge hints and solutions Challenge 48, page 64: Despite a huge number of attempts and the promise of eternal fame, this is the sober conclusion. Challenge 53, page 68: Yes, the argument is correct. In fact, more detailed discussions show that classical electrodynamics is in contradiction with all colours observed in nature. Ref. 178 Challenge 57, page 73: The calculation is not easy, but not too difficult either. For an initial ori- entation close to the vertical, the fall time 𝑇 turns out to be 1 8 𝑇= 𝑇 ln (154) 2π 0 𝛼 where 𝛼 is the starting angle, and a fall through angle π is assumed. Here 𝑇0 is the oscillation time of the pencil for small angles. (Can you determine it?) The indeterminacy relation for the tip of the pencil yields a minimum starting angle, because the momentum indeterminacy cannot be made arbitrarily large. You should be able to provide an upper limit. Once this angle is known, you can calculate the maximum time. Challenge 58, page 74: Use the temperature to calculate the average kinetic energy, and thus the average speed of atoms. Motion Mountain – The Adventure of Physics Challenge 59, page 74: At such low temperatures, the atoms cannot be fully distinguished; they form a state of matter with peculiar properties, called a condensate. The condensate is not at rest either; but due to its large mass, its fluctuations are greatly reduced, compared to those of a single atom. Challenge 61, page 78: Only variables whose product has the same units as physical action – Js – can be complementary to each other. Challenge 62, page 79: Use Δ𝐸 < 𝐸 and 𝑎 Δ𝑡 < 𝑐. Challenge 67, page 86: The quantum of action does not apply only to measurements, it applies to motion itself, and in particular, to all motion. Also effects of the nuclear forces, of nuclear copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net particles and of nuclear radiation particles must comply to the limit. And experiments show that they indeed do. In fact, if they did not, the quantum of action in electrodynamic situations could be circumvented, as you can check. Challenge 74, page 96: Outside the garage, all atoms need to form the same solid structure again. Challenge 75, page 97: Terabyte chips would need to have small memory cells. Small cells imply thin barriers. Thin barriers imply high probabilities for tunnelling. Tunnelling implies lack of memory. Challenge 81, page 108: If a particle were not elementary, its components would be bound by an interaction. But there are no known interactions outside those of the standard model. Challenge 82, page 109: The difficulties to see hydrogen atoms are due to their small size and their small number of electrons. As a result, hydrogen atoms produce only weak contrasts in X- ray images. For the same reasons it is difficult to image them using electrons; the Bohr radius of hydrogen is only slightly larger than the electron Compton wavelength. For the first time, in 2008, a research team claimed to have imaged hydrogen atoms adsorbed on graphene with the help of a transmission electron microscope. For details, see J. C. Meyer, C. O. Grit, M. F. Crommle & A. Zetti, Imaging and dynamics of light atoms and molecules on graphene, Nature 454, pp. 319–322, 2008. However, it seems that the report has not been con- firmed by another group yet. More hydrogen images have appeared in recent years. You may search for olympicene on the Page 185 internet, for example. For another recent result about hydrogen imaging, see above. Challenge 84, page 109: This is not easy! Can you use the concept of action to show that there indeed is a fundamental difference between very similar and very different operators? challenge hints and solutions 243 Challenge 86, page 110: 𝑟 = 86 pm, thus 𝑇 = 12 eV. This compares to the actual value of 13.6 eV. The trick for the derivation of the formula is to use ⟨𝜓 | 𝑟𝑥2 | 𝜓⟩ = 13 ⟨𝜓 | 𝑟𝑟 | 𝜓⟩, a relation valid for states with no orbital angular momentum. It is valid for all coordinates and also for the three momentum observables, as long as the system is non-relativistic. Challenge 87, page 111: A quantum fluctuation would require the universe to exist already. Such statements, regularly found in the press, are utter nonsense. Challenge 88, page 112: Point particles cannot be marked; nearby point particles cannot be dis- tinguished, due to the quantum of action. Challenge 89, page 112: The solution is two gloves. In the original setting, if two men and two women want to make love without danger, in theory they need only two condoms. Challenge 94, page 114: The Sackur–Tetrode formula is best deduced in the following way. We start with an ideal monoatomic gas of volume 𝑉, with 𝑁 particles, and total energy 𝑈. In phase space, state sum 𝑍 is given by 𝑉𝑁 1 𝑍= . (155) 𝑁! Λ3𝑁 Motion Mountain – The Adventure of Physics We use Stirling’s approximation 𝑁! ≈ 𝑁𝑁 /𝑒𝑁 , and the definition of the entropy as 𝑆 = ∂(𝑘𝑇 ln 𝑍)/∂𝑇. Inserting the definition of Λ, this gives the Sackur–Tetrode equation. Challenge 96, page 117: To write anything about two particles on paper, we need to distinguish them, even if the distinction is arbitrary. Challenge 99, page 123: The idea, also called quantum money, is not compatible with the size and lifetime requirements of actual banknotes. Challenge 100, page 124: Twins differ in the way their intestines are folded, in the lines of their hands and other skin folds. Sometimes, but not always, features like black points on the skin are mirror inverted on the two twins. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Challenge 109, page 135: Three. Challenge 110, page 135: Not for a mattress. This is not easy to picture. Challenge 111, page 136: Angels can be distinguished by name, can talk and can sing; thus they are made of a large number of fermions. In fact, many angels are human sized, so that they do not even fit on the tip of a pin. Challenge 117, page 140: A boson can be represented by an object glued to one infinitesimally thin thread whose two tails reach spatial infinity. Challenge 118, page 141: Trees, like all macroscopic objects, have a spin value that depends on their angular momentum. Being classical objects whose phase can be observed, the spin value is uncertain. It makes no sense to ask whether trees or other macroscopic objects are bosons or fermions, as they are not quantons. Challenge 121, page 142: Ghosts, like angels, can be distinguished by name, can talk and can be seen; thus they contain fermions. However, they can pass through walls and they are trans- parent; thus they cannot be made of fermions, but must be images, made of bosons. That is a contradiction. Challenge 122, page 144: Macroscopic superpositions cannot be drawn, because observation implies interaction with a bath, which destroys macroscopic superposition. Challenge 124, page 146: The loss of non-diagonal elements leads to an increase in the diagonal elements, and thus of entropy. Challenge 127, page 153: The energy speed is given by the advancement of the outer two tails; that speed is never larger than the speed of light. 244 challenge hints and solutions Challenge 128, page 155: No, as taking a photo implies an interaction with a bath, which would destroy the superposition. In more detail, a photograph requires illumination; illumination is a macroscopic electromagnetic field; a macroscopic field is a bath; a bath implies decoherence; decoherence destroys superpositions. Challenge 131, page 157: It depends. They can be due to interference or to intensity sums. In the case of radio the effect is clearer. If at a particular frequency the signals changes periodically from one station to another, one has a genuine interference effect. Challenge 132, page 157: They interfere. But this is a trick question; what is a monochromatic electron? Does it occur in the laboratory? Challenge 133, page 157: Such a computer requires clear phase relations between components; such phase relations are extremely sensitive to outside disturbances. At present, they do not hold longer than a hundred microseconds, whereas long computer programs require minutes and hours to run. Challenge 134, page 157: A record is an effect of a process that must be hard to reverse or undo. The traces of a broken egg are easy to clean on a large glass plate, but hard in the wool of a sheep. Motion Mountain – The Adventure of Physics Broken teeth, torn clothes, or scratches on large surfaces are good records. Forensic scientists know many additional examples. Challenge 138, page 166: Any other bath also does the trick, such as the atmosphere, sound vi- brations, electromagnetic fields, etc. Challenge 139, page 166: The Moon is in contact with baths like the solar wind, falling meteor- ites, the electromagnetic background radiation of the deep universe, the neutrino flux from the Sun, cosmic radiation, etc. Challenge 140, page 168: Spatially periodic potentials have the property. Decoherence then leads to momentum diagonalization. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Challenge 142, page 171: If so, let the author know. Challenge 143, page 182: The red shift value is 𝑧 = 9.9995. From the formula for the longitudinal Doppler shift we get 𝑣/𝑐 = ((𝑧 + 1)2 − 1)/((𝑧 + 1)2 + 1); this yields 0.984 in the present case. The galaxy thus moves away from Earth with 98.4 % of the speed of light. Challenge 149, page 184: Hydrogen atoms are in eigenstates for the reasons explained in the chapter on superpositions and probabilities: in a gas, atoms are part of a bath, and thus almost always in energy eigenstates. Challenge 154, page 195: If several light beams are focused in the space between the mirrors, and if the light beam frequency is properly tuned with respect to the absorption frequencies of the atoms, atoms will experience a restoring force whenever they move away from the focus region. By shining light beams to the focus region from 6 directions, atoms are trapped. The technique of laser cooling is now widely used in research laboratories. Challenge 155, page 196: No, despite its name, phosphorus is not phosphorescent, but chemo- luminescent. Challenge 157, page 197: This is a trick question. A change in 𝛼 requires a change in 𝑐, ℏ, 𝑒 or 𝜀0 . None of these changes is possible or observable, as all our measurement apparatus are based on these units. Speculations about change of 𝛼, despite their frequency in the press and in scientific journals, are idle talk. Challenge 158, page 197: A change of physical units such that ℏ = 𝑐 = 𝑒 = 1 would change the value of 𝜀0 in such a way that 4π𝜀o = 1/𝛼 ≈ 137.036. Challenge 161, page 207: Mass is a measure of the amount of energy. The ‘square of mass’ makes no sense. challenge hints and solutions 245 Challenge 165, page 210: Planck limits can be exceeded for extensive observables for which many particle systems can exceed single particle limits, such as mass, momentum, energy or electrical resistance. Challenge 167, page 212: Do not forget the relativistic time dilation. Challenge 168, page 212: The formula with 𝑛 − 1 is a better fit. Why? Challenge 171, page 213: No! They are much too precise to make sense. They are only given as an illustration for the behaviour of the Gaussian distribution. Real measurement distributions are not Gaussian to the precision implied in these numbers. Challenge 172, page 213: About 0.3 m/s. It is not 0.33 m/s, it is not 0.333 m/s and it is not any longer strings of threes. Challenge 174, page 219: The slowdown goes quadratically with time, because every new slow- down adds to the old one! Challenge 175, page 220: No, only properties of parts of the universe are listed. The universe Vol. VI, page 112 itself has no properties, as shown in the last volume. Challenge 176, page 221: The double of that number, the number made of the sequence of all Motion Mountain – The Adventure of Physics even numbers, etc. Challenge 179, page 225: We will find out in the last volume that all measurement values have upper and lower bounds. We will also find out that two physical measurement results cannot differ just from, say, the 300th decimal place onwards. So indeed, all measurement results are real numbers, but not vice versa. It needs to be stressed that for quantum theory, for relativity and also for Galilean physics this restriction has no consequences whatsoever. 𝑎 𝑏 Challenge 181, page 226: |𝑧|2 is the determinant of the matrix 𝑧 = ( ). −𝑏 𝑎 Challenge 186, page 227: Use Cantor’s diagonal argument, as in challenge 274. Challenge 187, page 228: Any quaternion 𝑋 = 𝑎𝑖+𝑏𝑗+𝑐𝑘 with 𝑎2 +𝑏2 +𝑐2 = 1 solves the equation copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 𝑋2 + 1 = 0; the purely imaginary solutions +𝑖 and −𝑖 are thus augmented by a continuous sphere of solutions in quaternion space. Challenge 190, page 230: Any rotation by an angle 2π is described by −1. Only a rotation by 4π is described by +1; quaternions indeed describe spinors. Challenge 192, page 232: Just check the result component by component. See also the men- tioned reference. Challenge 194, page 234: No. Because the unit octonions are not associative, they do not form a group at all. Despite its superficial appeal, this line of reasoning has not led to any insight into the nature of the fundamental interactions. Challenge 195, page 235: For a Gaussian integer 𝑛 + 𝑖𝑚 to be prime, the integer 𝑛2 + 𝑚2 must be prime, and in addition, a condition on 𝑛 mod 3 must be satisfied; which one and why? Challenge 197, page 236: The set that contains only the zero vector. Challenge 198, page 236: The metric is regular, positive definite and obeys the triangle inequal- ity. Challenge 200, page 237: Essentially only the vector spaces listed in the appendix (or in the book). Challenge 201, page 237: If you cannot, blame your math teacher at secondary school, and then look up the definitions. It is not a difficult topic. Challenge 202, page 238: Spaces could exist approximately, as averages of non-continuous structures. This idea is explored in modern research; an example is given in the last volume of this series. Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net challenge hints and solutions 246 BI BL IO G R A PH Y “ No man but a blockhead ever wrote except for ” money. Samuel Johnson “ ” As soon as you write, no time to read remains. Anonymous Motion Mountain – The Adventure of Physics 1 Giuseppe Fumagalli, Chi l’ha detto?, Hoepli, Milano, 1983. Cited on page 15. 2 The quantum of action was introduced in Max Planck, Über irreversible Strahlungs- vorgänge, Sitzungsberichte der Preußischen Akademie der Wissenschaften, Berlin pp. 440– 480, 1899. In the paper, Planck used the letter 𝑏 for what nowadays is called ℎ. Cited on page 17. 3 Bohr explained the indivisibilty of the quantum of action in his famous Como lecture. See N. B ohr, Atomtheorie und Naturbeschreibung, Springer, 1931. On page 16 he writes: ‘No more is it likely that the fundamental concepts of the classical theories will ever become superfluous for the description of physical experience. The recognition of the indivisibility copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net of the quantum of action, and the determination of its magnitude, not only depend on an analysis of measurements based on classical concepts, but it continues to be the applica- tion of these concepts alone that makes it possible to relate the symbolism of the quantum theory to the data of experience.’ He also writes: ‘...the fundamental postulate of the in- divisibility of the quantum of action is itself, from the classical point of view, an irrational element which inevitably requires us to forgo a causal mode of description and which, be- cause of the coupling between phenomena and their observation, forces us to adopt a new mode of description designated as complementary in the sense that any given application of classical concepts precludes the simultaneous use of other classical concepts which in a different connection are equally necessary for the elucidation of the phenomena ...’ and ‘...the finite magnitude of the quantum of action prevents altogether a sharp distinction being made between a phenomenon and the agency by which it is observed, a distinction which underlies the customary concept of observation and, therefore, forms the basis of the classical ideas of motion.’ Other statements about the indivisibility of the quantum of action can be found in N. B ohr, Atomic Physics and Human Knowledge, Science Editions, 1961. See also Max Jammer, The Philosophy of Quantum Mechanics, Wiley, first edition, 1974, pp. 90–91. Cited on page 17. 4 For some of the rare modern publications emphasizing the quantum of action see M. B. Mensky, The action uncertainty principle and quantum gravity, Physics Letters A 162, p. 219, 1992, and M. B. Mensky, The action uncertainty principle in continuous quantum measurements, Physics Letters A 155, pp. 229–235, 1991. Schwinger’s quantum- action principle is also used in Richard F. W. Bader, Atoms in Molecules – A Quantum Theory, Oxford University Press, 1994. 248 bibliography There is a large number of general textbooks on quantum theory. There is one for every taste. A well-known conceptual introduction is Jean-Marc Lév y-Leblond & Françoise Balibar, Quantique – Rudiments, Masson, 1997, translated into English as Quantics, North-Holland, 1990. One of the most beautiful books is Julian Schwinger, Quantum Mechanics – Sym- bolism of Atomic Measurements, edited by Berthold-Georg Englert, Springer Verlag, 2001. A modern approach with a beautiful introduction is Max Schubert & Ger- hard Weber, Quantentheorie – Grundlagen und Anwendungen, Spektrum Akademischer Verlag, 1993. A standard beginner’s text is C. Cohen-Tannoudji, B. Diu & F. Laloë, Méca- nique quantique I et II, Hermann, Paris, 1977. It is also available in several translations. A good text is Asher Peres, Quantum Theory – Concepts and Methods, Kluwer, 1995. For a lively approach, see Vincent Icke, The Force of Symmetry, Cambridge Univer- sity Press, 1994. New textbooks are published regularly around the world. Cited on pages 17 and 255. Motion Mountain – The Adventure of Physics 5 The best source for the story about the walk in the forest with Planck’s son Erwin is Hans Roos & Armin Hermann, editors, Max Planck – Vorträge, Reden, Erinnerungen, Springer, 2001, page 125. As the text explains, the story was told by Erwin Planck to at least two different people. Erwin Planck himself was part of the failed 1944 plot against Hitler and was hanged in January 1945. Cited on page 20. 6 Max B orn, Zur Quantenmechanik der Stoßvorgänge (vorläufige Mitteilung), Zeitschrift für Physik 37, pp. 863–867, 1926, Max B orn, Quantenmechanik der Stoßvorgänge, Zeits- chrift für Physik 38, pp. 803–827, 1926. Cited on page 24. 7 See for example the papers by Jan Hilgevoord, The uncertainty principle for energy and time, American Journal of Physics 64, pp. 1451–1456, 1996, and by Paul Busch, On the copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net time–energy uncertainty reaction, parts 1 & 2, Foundations of Physics 20, pp. 1–43, 1990. A classic is the paper by Eugene P. Wigner, On the time–energy uncertainty relation, in Abdus Salam & Eugene P. Wigner, editors, Aspects of Quantum Theory, Cambridge University Press, 1972. Cited on page 25. 8 See also the booklet by Claus Mattheck, Warum alles kaputt geht – Form und Versagen in Natur und Technik, Forschungszentrum Karlsruhe, 2003. Cited on page 30. 9 R. Clifton, J. Bub & H. Halvorson, Characterizing quantum theory in terms of information-theoretic constraints, arxiv.org/abs/quant-ph/0211089. Cited on page 36. 10 This way to look at cans of beans goes back to the text by Susan Hewitt & Ed- ward Subitzky, A call for more scientific truth in product warning labels, Journal of Ir- reproducible Results 36, nr. 1, 1991. Cited on page 37. 11 J. Malik, The yields of the Hiroshima and Nagasaki nuclear explosions, Technical Report LA-8819, Los Alamos National Laboratory, September 1985. Cited on page 38. 12 The quotes on motion are found in chapter VI of F. Engels, Herrn Eugen Dührings Um- wälzung der Wissenschaft, Verlag für fremdsprachliche Literatur, 1946. The book is com- monly called Anti-Dühring. Cited on pages 39 and 74. 13 Rodney Loudon, The Quantum Theory of Light, Oxford University Press, 2000. Cited on page 40. 14 E. M. Brumberg & S. I. Vavilov, Izvest. Akad. Nauk. Omen Ser. 7, p. 919, 1933. Cited on page 40. 15 On photon detection in the human eye, see the influential review by F. Rieke & bibliography 249 D. A. Baylor, Single-photon detection by rod cells of the retina, Reviews of Modern Physics 70, pp. 1027–1036, 1998. It can be found on the internet as a pdf file. Cited on page 42. 16 F. Rieke & D. A. Baylor, Single-photon detection by rod cells of the retina, Reviews of Modern Physics 70, pp. 1027–1036, 1998. They also mention that the eye usually works at photon fluxes between 108 /𝜇m2 s (sunlight) and 10−2 /𝜇m2 s (starlight). The cones, in the retina detect, in colour, light intensities in the uppermost seven or eight decades, whereas the rods detect, in black and white, the lower light intensities. Cited on page 44. 17 E. Fischbach, H. Kloor, R. A. Langel, A. T. Y. Lui & M. Peredo, New geomag- netic limit on the photon mass and on long-range forces coexisting with electromagnetism, Physical Review Letters 73, pp. 514–517, 1994. Cited on page 45. 18 A. H. Compton, The scattering of X-rays as particles, American Journal of Physics 29, pp. 817–820, 1961. This is a pedagogical presentation of the discoveries he made in 1923. Cited on page 45. 19 The reference paper on this topic is G. Breitenbach, S. Schiller & J. Mlynek, Motion Mountain – The Adventure of Physics Measurement of the quantum states of squeezed light, 387, pp. 471–475, 1997. It is available freely at gerdbreitenbach.de/publications/nature1997.pdf. Cited on pages 48 and 241. 20 The famous paper is R. Hanbury Brown & R. Q. Twiss, Nature 178, p. 1046, 1956. They got the idea to measure light in this way from their earlier work, which used the same method with radio waves: R. Hanbury Brown & R. Q. Twiss, Nature 177, p. 27, 1956. The complete discussion is given in their papers R. Hanbury Brown & R. Q. Twiss, Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation, Proceedings of the Royal Society A 242, pp. 300– 324, 1957, and R. Hanbury Brown & R. Q. Twiss, Interferometry of the intensity fluc- tuations in light. II. An experimental test of the theory for partially coherent light, Proceedings copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net of the Royal Society A 243, pp. 291–319, 1958. Both are dowloadable for free on the internet and are well worth reading. Cited on page 52. 21 J. Glanz, First light from a space laser, Science 269, p. 1336, 1995. Cited on page 54. 22 A. Einstein, Über einen die Erzeugung und Umwandlung des Lichtes betreffenden heur- istischen Standpunkt, Annalen der Physik 17, pp. 132–184, 1905. Cited on page 55. 23 See the summary by P. W. Milonni, Answer to question 45: What (if anything) does the photoelectric effect teach us?, American Journal of Physics 65, pp. 11–12, 1997. Cited on page 55. 24 For a detailed account, See J. J. Prentis, Poincaré ’s proof of the quantum discontinu- ity of nature, American Journal of Physics 63, pp. 339–350, 1995. The original papers are Henri Poincaré, Sur la théorie des quanta, Comptes Rendus de l’Académie des Sci- ences (Paris) 153, pp. 1103–1108, 1911, and Henri Poincaré, Sur la théorie des quanta, Journal de Physique (Paris) 2, pp. 5–34, 1912. Cited on page 55. 25 J. Jacobson, G. Björk, I. Chang & Y. Yamamoto, Photonic de Broglie waves, Physical Review Letters 74, pp. 4835–4838, 1995. The first measurement was published by E. J. S. Fonseca, C. H. Monken & S. de Pádua, Measurement of the de Broglie wavelength of a multiphoton wave packet, Physical Review Letters 82, pp. 2868–2671, 1995. Cited on page 55. 26 For the three-photon state, see M. W. Mitchell, J. S. Lundeen & A. M. Steinberg, Super-resolving phase measurements with a multiphoton entangled state, Nature 429, pp. 161– 164, 2004, and for the four-photon state see, in the same edition, P. Walther, J. -W. Pan, 250 bibliography M. Aspelmeyer, R. Ursin, S. Gasparoni & A. Zeilinger, De Broglie wavelength of a non-local four-photon state, Nature 429, pp. 158–161, 2004. Cited on page 55. 27 For an introduction to squeezed light, see L. Mandel, Non-classical states of the electro- magnetic field, Physica Scripta T 12, pp. 34–42, 1986. Cited on page 55. 28 Friedrich Herneck, Einstein und sein Weltbild: Aufsätze und Vorträge, Buchverlag Der Morgen, 1976, page 97. Cited on page 56. 29 The famous quote on single-photon interference is found on page 9 of famous, beautiful but difficult textbook P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, 1930. It is also discussed, in a somewhat confused way, in the otherwise informative article by H. Paul, Interference between independent photons, Reviews of Modern Physics 58, pp. 209–231, 1986. Cited on pages 59 and 66. 30 The original papers on coherent states are three: R. J. Glauber, The quantum theory of optical coherence, Physical Review 130, pp. 2529–2539, 1963, J. R. Klauder, Continuous- representation theory, I and II, Journal of Mathematical Physics 4, pp. 1055–1058, 1963, and E. C. G. Sudarshan, Equivalence of semiclassical and quantum mechanical descriptions Motion Mountain – The Adventure of Physics of statistical light beams, Physical Review Letters 10, p. 227, 1963. Cited on page 63. 31 See, for example the wonderful text Richard P. Feynman, QED – The Strange Theory of Light and Matter, pp. 73–75, Princeton University Press, 1988, or Richard P. Feynman & Steven Weinberg, Elementary Particles and the Laws of Physics, p. 23, Cambridge University Press 1987. Cited on page 63. 32 Wolf gang Tittel, J. Brendel, H. Zbinden & N. Gisin, Violation of Bell inequal- ities by photons more than 10 km apart, Physical Review Letters 81, pp. 3563–3566, 26 Oc- tober 1998. Cited on page 64. 33 N. B ohr & L. Rosenfeld, Zur Frage der Meßbarkeit der elektromagnetischen Feld- größen, Mat.-fys. Medd. Danske Vid. Selsk. 12, p. 8, 1933. The results were later published in copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net English as N. B ohr & L. Rosenfeld, Field and charge measurements in quantum elec- trodynamics, Physical Review 78, pp. 794–798, 1950. Cited on page 65. 34 Misleading statements are given in the introduction and in the conclusion of the review by H. Paul, Interference between independent photons, Review of Modern Physics 58, pp. 209–231, 1986. However, in the bulk of the article the author in practice retracts the statement, e.g. on page 221. Cited on page 66. 35 G. Magyar & L. Mandel, Interference fringes produced by superposition of two inde- pendent maser light beams, Nature 198, pp. 255–256, 1963. Cited on page 67. 36 R. Kidd, J. Aedini & A. Anton, Evolution of the modern photon, American Journal of Physics 57, pp. 27–35, 1989, Cited on page 69. 37 The whole bunch of atoms behaves as one single molecule; one speaks of a Bose–Einstein condensate. The first observations, worthy of a Nobel prize, were by M.H. Ander- son & al., Observation of Bose–Einstein condensation in a dilute atomic vapour, Science 269, pp. 198–201, 1995, C. C. Bradley, C. A. Sackett, J. J. Tollett & R. G. Hulet, Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions, Physical Review Letters 75, pp. 1687–1690, 1995, K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn & W. Ketterle, Bose–Einstein con- densation in a gas of sodium atoms, Physical Review Letters 75, pp. 3969–3973, 1995. For a simple introduction, see W. Ketterle, Experimental studies of Bose–Einstein condensa- tion, Physics Today pp. 30–35, December 1999. Cited on page 74. 38 J. L. Costa-Krämer, N. Garcia, P. García-Mochales & P. A. Serena, Nanowire formation in macroscopic metallic contacts: a universal property of metals, Surface bibliography 251 Science Letters 342, pp. L1144–L1152, 1995. See also J. L. Costa-Krämer, N. Garcia, P. A. Serena, P. García-Mochales, M. Marqués & A. Correia, Conductance quantization in nanowires formed in macroscopic contacts, Physical Review B p. 4416, 1997. Cited on page 74. 39 The beautiful undergraduate experiments made possible by this discovery are desribed in E. L. Foley, D. Candela, K. M. Martini & M. T. Tuominen, An undergradu- ate laboratory experiment on quantized conductance in nanocontacts, American Journal of Physics 67, pp. 389–393, 1999. Cited on pages 74 and 75. 40 L. de Broglie, Ondes et quanta, Comptes rendus de l’Académie des Sciences 177, pp. 507–510, 1923. Cited on page 76. 41 C. Jönsson, Interferenz von Elektronen am Doppelspalt, Zeitschrift für Physik 161, pp. 454–474, 1961, C. Jönsson, Electron diffraction at multiple slits, American Journal of Physics 42, pp. 4–11, 1974. Because of the charge of electons, this experiment is not easy to perform: any parts of the set-up that are insulating get charged and distort the picture. That is why the experient was performed much later with electrons than with atoms, neutrons and molecules. Cited on page 77. Motion Mountain – The Adventure of Physics 42 M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw & A. Zeilinger, Wave–particle duality of C60 molecules, Nature 401, pp. 680–682, 14 October 1999. See also the observation for tetraphenyleprophyrin and C60 F48 by the same team, published as L. Hackermüller & al., Wave nature of biomolecules and fluorofullerenes, Physical Review Letters 91, p. 090408, 2003. No phenomoenon of quantum theory has been experimentally studied as much as quantum interference. The transition from interference to non-interference has also been explored, as in P. Facchi, A. Mariano & S. Pascazio, Mesoscopic interference, Re- cent Developments in Physics 3, pp. 1–29, 2002. Cited on page 77. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 43 G. Papini, Shadows of a maximal acceleration, arxiv.org/abs/gr-qc/0211011. Cited on page 79. 44 J. Perrin, Nobel Prize speech, found at www.nobel.se, and H. Nagaoka, Kinetics of a system of particles illustrating the line and the band spectrum and the phenomena of radio- activity, Philosophical Magazine S6, 7, pp. 445–455, March 1904. Cited on page 79. 45 N. 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Padgett, Uncertainty principle for angular position and angular momentum, New Journal of Physics 6, p. 103, 2004. This is a freely accessible online journal. Cited on page 82. 50 W. Gerlach & O. Stern, Der experimentelle Nachweis des magnetischen Moments des Silberatoms, Zeitschrift für Physik 8, p. 110, 1921. See also the pedagogical explanation by 252 bibliography M. Hannout, S. Hoyt, A. Kryowonos & A. Widom, Quantum measurement and the Stern–Gerlach experiment, American Journal of Physics 66, pp. 377–379, 1995. Cited on page 83. 51 J. P. Woerdman, G. Nienhuis, I. Kuščer, Is it possible to rotate an atom?, Op- tics Communications 93, pp. 135–144, 1992. We are talking about atoms rotating around their centre of mass; atoms can of course rotate around other bodies, as discussed by M. P. Silverman, Circular birefringence of an atom in uniform rotation: the classical per- spective, American Journal of Physics 58, pp. 310–317, 1990. Cited on page 85. 52 J. Schmiedmayer, M. S. Chapman, C. R. Ekstrom, T. D. Hammond, S. Wehinger & D. E. Pritchard, Index of refraction of various gases for sodium matter waves, Physical Review Letters 74, p. 1043-1046, 1995. Cited on page 85. 53 The original result is due to V. de Sabbata & C. Sivaram, A minimal time and time- temperature uncertainty principle, Foundations of Physics Letters 5, pp. 183–189, 1992. Ex- perimental details are found, for example, in G. T. Gillies & S. W. Allison, Experi- mental test of a time-temperature formulation of the uncertainty principle via nanoparticle Motion Mountain – The Adventure of Physics fluorescence, Foundations of Physics Letters 18, pp. 65–74, 2005. Cited on page 86. 54 Albert Einstein & Max B orn, Briefwechsel 1916 bis 1955, Rowohlt, 1969, as cited on page 34. Cited on page 87. 55 E. Schrödinger, Quantisierung als Eigenwertproblem I, Annalen der Physik 79, pp. 361– 376, 1926, and Quantisierung als Eigenwertproblem II, Annalen der Physik 79, pp. 489–527, 1926. Cited on page 92. 56 C. G. 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Cited on page 101. 61 G. E. Uhlenbeck & S. Goudsmit, Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons, Naturwissenschaften 13, pp. 953–954, 1925. Cited on page 104. 62 L. Thomas, The motion of the spinning electron, Nature 117, p. 514, 1926. Cited on page 105. 63 K. von Meyenn & E. Schucking, Wolfgang Pauli, Physics Today pp. 43–48, February 2001. Cited on page 105. 64 T. D. Newton & E. P. Wigner, Localized states for elementary systems, Review of Mod- ern Physics 21, pp. 400–406, 1949. L. L. Foldy & S. A. Wouthuysen, On the Dirac theory of spin 1/2 particles and its nonrelativistic limit, Physical Review 78, pp. 29–36, 1950. Both are classic papers. Cited on page 106. bibliography 253 65 J. P. Costella & B. H. J. McKellar, The Foldy–Wouthuysen transformation, Amer- ican Journal of Physics 63, pp. 1119–1121, 1995. Cited on page 106. 66 For an account of the first measuremnt of the g-factor of the electron, see H. R. Crane, How we happended to measure g-2: a tale of serendipity, Physics in Perspective 2, pp. 135–140, 2000. The most interesting part is how the experimentalists had to overcome the conviction of almost all theorists that the measurement was impossible in principle. Cited on page 107. 67 The 𝑔-factors for composite nuclei are explained briefly on en.wikipedia.org/wiki/ Nuclear_magnetic_moment and measured values are found at www-nds.iaea.org. See also H. Dehmelt, Is the electron a composite particle?, Hyperfine Interactions 81, pp. 1–3, 1993. Cited on pages 108 and 254. 68 The nearest anyone has come to an image of a hydrogen atom is found in A. Yazdani, Watching an atom tunnel, Nature 409, pp. 471–472, 2001. The experiments on Bose–Einstein condensates are also candidates for images of hydrogen atoms. The company Hitachi made a fool of itself in 1992 by claiming in a press release that its newest electron microscope could image hydrogen atoms. Cited on page 109. A. M. Wolsky, Kinetic energy, size, and the uncertainty principle, American Journal of Motion Mountain – The Adventure of Physics 69 Physics 42, pp. 760–763, 1974. Cited on page 110. 70 For a fascinating summary, see M. A. Cirone, G. Metikas & W. P. Schleich, Un- usual bound or localized states, preprint at arxiv.org/abs/quant-ph/0102065. Cited on page 110. 71 See the paper by Martin Gardner, Science fiction puzzle tales, Clarkson Potter, 67, pp. 104–105, 1981, or his book Aha! Insight, Scientific American & W.H. Freeman, 1978. Sev- eral versions are given A. Hajnal & P. Lovász, An algorithm to prevent the propagation of certain diseases at minimum cost, in Interfaces Between Computer Science and Operations Research, edited by J. K. Lenstra, A. H. G. Rinnooy Kan & P. Van Emde B oas, copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Mathematisch Centrum, Amsterdam 1978, whereas the computer euphemism is used by A. Orlitzky & L. Shepp, On curbing virus propagation, Technical memorandum, Bell Labs 1989. Cited on page 112. 72 A complete discussion of the problem can be found in chapter 10 of Ilan Vardi, Compu- tational Recreations in Mathematica, Addison Wesley, 1991. Cited on page 112. 73 On Gibbs’ paradox, see your favourite text on thermodynamics or statistical mechanics. See also W. H. Zurek, Algorithmic randomness and physical entropy, Physical Review A 40, pp. 4731–4751, 1989. Zurek shows that the Sackur–Tetrode formula can be derived from algorithmic entropy considerations. Cited on page 114. 74 S. N. B ose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik 26, pp. 178– 181, 1924. The theory was then expanded in A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin 22, pp. 261–267, 1924, A. Einstein, Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin 23, pp. 3–14, 1925, A. Einstein, Zur Quantentheorie des idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin 23, pp. 18–25, 1925. Cited on page 118. 75 C. K. Hong, Z. Y. Ou & L. Mandel, Measurement of subpicosecond time inter- vals between two photons by interference, Physical Review Letters 59, pp. 2044–2046, 1987. See also T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko & Y. H. Shih, Can two-photon interference be considered the inter- ference of two photons?, Physical Review Letters 77, pp. 1917–1920, 1996. Cited on page 118. 254 bibliography 76 An example of such an experiment performed with electrons instead of photons is described in E. B ocquillon, V. Freulon, J. -M. Berroir, P. Degiovanni, B. Plaçais, A. Cavanna, Y. Jin & G. Fève, Coherence and indistinguishability of single electrons emitted by independent sources, Science 339, pp. 1054–1057, 2013. See also the comment C. Schönenberger, Two indistinguishable electrons interfere in an electronic device, Science 339, pp. 1041–1042, 2013. Cited on page 119. 77 M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. B oiron, C. I. Westbrook & A. Aspect, Hanbury Brown Twiss effect for ultracold quantum gases, Science 310, p. 648, 2005, preprint at arxiv.org/abs/cond-mat/0508466. J. Viana Gomes, A. Perrin, M. Schellekens, D. B oiron, C. I. Westbrook & M. Belsley, Theory for a Hanbury Brown Twiss experiment with a ballistically expand- ing cloud of cold atoms, Physical Review A 74, p. 053607, 2006, preprint at arxiv.org/ abs/quant-ph/0606147. T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. B oiron, A. Aspect & C. I. Westbrook, Comparison of the Hanbury Brown-Twiss effect for bo- sons and fermions, Nature 445, p. 402, 2007, preprint at arxiv.org/abs/cond-mat/0612278. Motion Mountain – The Adventure of Physics Cited on page 120. 78 The experiment is described in E. Ramberg & G. A. Snow, Experimental limit on a small violation of the Pauli principle, Physics Letters B 238, pp. 438–441, 1990. Other experimental tests are reviewed in O. W. Greenberg, Particles with small violations of Fermi or Bose statistics, Physical Review D 43, pp. 4111–4120, 1991. Cited on page 122. 79 The original no-cloning theorem is by D. Dieks, Communication by EPR devices, Phys- ics Letters A 92, pp. 271–272, 1982, and by W. K. Wootters & W. H. Zurek, A single quantum cannot be cloned, Nature 299, pp. 802–803, 1982. For a discussion of photon and multiparticle cloning, see N. Gisin & S. Massar, Optimal quantum cloning machines, Physics Review Letters 79, pp. 2153–2156, 1997. The whole topic has been presented in detail copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net by V. Buzek & M. Hillery, Quantum cloning, Physics World 14, pp. 25–29, November 2001. Cited on page 122. 80 S. J. Wiesner, Conjugate Coding, SIGACT News, 15, pp. 78–88, 1983. This widely cited paper was one of starting points of quantum information theory. Cited on page 123. 81 The most recent experimental and theoretical results on physical cloning are described in A. Lamas-Linares, C. Simon, J. C. Howell & D. B ouwmeester, Experi- mental quantum cloning of single photons, Science 296, pp. 712 – 714, 2002, D. Collins & S. Popescu, A classical analogue of entanglement, preprint arxiv.org/abs/quant-ph/ 0107082, 2001, and A. Daffertshofer, A. R. Plastino & A. Plastino, Classical no-cloning theorem, Physical Review Letters 88, p. 210601, 2002. Cited on page 123. 82 E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals of Mathematics 40, pp. 149–204, 1939. This famous paper summarises the work which later brought him the Nobel Prize in Physics. Cited on pages 125 and 137. 83 For a full list of isotopes, see R. B. Firestone, Table of Isotopes, Eighth Edition, 1999 Up- date, with CDROM, John Wiley & Sons, 1999. Cited on page 127. 84 This is deduced from the 𝑔 − 2 measurements, as explained in his Nobel-prize talk by Hans Dehmelt, Experiments with an isolated subatomic particle at rest, Reviews of Mod- ern Physics 62, pp. 525–530, 1990. On this topic, see also his paper Ref. 67. No citations. and in Hans Dehmelt, Is the electron a composite particle?, Hyperfine Interactions 81, pp. 1–3, 1993. 85 G. Gabrielse, H. Dehmelt & W. Kells, Observation of a relativistic, bistable hyster- esis in the cyclotron motion of a single electron, Physical Review Letters 54, pp. 537–540, bibliography 255 1985. No citations. 86 W. Pauli, The connection between spin and statistics, Physical Review 58, pp. 716– 722, 1940. Cited on page 133. 87 The belt trick has been popularized by Dirac, Feynman and many others. An example is R. P. Feynman, The reason for antiparticles, in Elementary Particles and the Laws of Physics: The 1986 Dirac Memorial Lectures, Cambridge University Press, 1987. The belt trick is also explained, for example, on page 1148 in C. W. Misner, K. S. Thorne & J. A. Wheeler, Gravitation, Freeman, 1973. It is called the scissor trick on page 43 of volume 1 of R. Penrose & W. Rindler, Spinors and Spacetime, 1984. It is also cited and discussed by R. Gould, Answer to question #7, American Journal of Physics 63, p. 109, 1995. Still, some physicists do not like the belt-trick image for spin 1/2 particles; for an example, see I. Duck & E. C. G. Sudarshan, Toward an understanding of the spin- statistics theorem, American Journal of Physics 66, pp. 284–303, 1998. Cited on page 133. 88 M. V. Berry & J. M. Robbins, Indistinguishability for quantum particles: spin, statist- ics and the geometric phase, Proceedings of the Royal Society in London A 453, pp. 1771– Motion Mountain – The Adventure of Physics 1790, 1997. See also the comments to this result by J. Twamley, Statistics given a spin, Nature 389, pp. 127–128, 11 September 1997. Their newer results are M. V. Berry & J. M. Robbins, Quantum indistinguishability: alternative constructions of the transpor- ted basis, Journal of Physics A (Letters) 33, pp. L207–L214, 2000, and M. V. Berry & J. M. Robbins, in Spin–Statistics, eds. R. Hilborn & G. Tino, American Institute of Physics, 2000, pp. 3–15. See also Michael Berry’s home page at www.phy.bris.ac.uk/people/ berry_mv. Cited on page 135. 89 R. W. Hartung, Pauli principle in Euclidean geometry, American Journal of Physics 47, pp. 900–910, 1979. Cited on page 136. 90 The issue is treated in his Summa Theologica, in question 52 of the first part. The com- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net plete text, several thousand pages, can be found on the www.newadvent.org website. Also present-day angelologists, of which there are only a few across the world, agree with Aqui- nas. Cited on page 136. 91 The point that spin can be seen as a rotation was already made by F. J. Belinfante, On the spin angular momentum of mesons, Physica 6, p. 887, 1939, and taken up again by Hans C. Ohanian, What is spin?, American Journal of Physics 54, pp. 500–505, 1986. See also E. Duran & A. Erschow, Physikalische Zeitschrift der Sowjetunion 12, p. 466, 1937. Cited on page 138. 92 See the book by Jean-Marc Lév y-Leblond & Françoise Balibar in Ref. 4. Cited on page 141. 93 Generalizations of bosons and fermions are reviewed in the (serious!) paper by O. W. Greenberg, D. M. Greenberger & T. V. Greenbergest, (Para)bosons, (para)fermions, quons and other beasts in the menagerie of particle statistics, at arxiv.org/ abs/hep-th/9306225. A newer summary is O. W. Greenberg, Theories of violation of statistics, electronic preprint available at arxiv.org/abs/hep-th/0007054. Cited on page 142. 94 Gell-Mann wrote this for the 1976 Nobel Conference (not for the Nobel speech; he is the only winner who never published it.) M. Gell-Mann, What are the building blocks of matter?, in D. Huff & O. Prewitt, editors, The Nature of the Physical Universe, New York, Wiley, 1979, p. 29. Cited on page 143. 95 See e.g. the reprints of his papers in the standard collection by John A. Wheeler & Wo- jciech H. Zurek, Quantum Theory and Measurement, Princeton University Press, 1983. Cited on page 144. 256 bibliography 96 H. D. Zeh, On the interpretation of measurement in quantum theory, Foundations of Phys- ics 1, pp. 69–76, 1970. Cited on page 144. 97 Linda Reichl, A Modern Course in Statistical Physics, Wiley, 2nd edition, 1998. An ex- cellent introduction into thermodynamics. Cited on page 146. 98 E. Joos & H. D. Zeh, The emergence of classical properties through interactions with the environment, Zeitschrift für Physik B 59, pp. 223–243, 1985. See also Erich Joos, Deco- herence and the appearance of a classical world in quantum theory, Springer Verlag, 2003. Cited on page 148. 99 M. Tegmark, Apparent wave function collapse caused by scattering, Foundation of Phys- ics Letters 6, pp. 571–590, 1993, preprint at arxiv.org/abs/gr-qc/9310032. See also his paper that shows that the brain is not a quantum computer, M. Tegmark, The importance of quantum decoherence in brain processes, Physical Review E 61, pp. 4194–4206, 2000, pre- print at arxiv.org/abs/quant-ph/9907009. Cited on page 148. 100 The decoherence time is bound from above by the relaxation time. See A. O. Caldeira & A. J. Leggett, Influence of damping on quantum interference: an exactly soluble model, Physical Review A 31, 1985, pp. 1059–1066. This is the main reference about effects of deco- Motion Mountain – The Adventure of Physics herence for a harmonic oscillator. The general approach to relate decoherence to the influ- ence of the environment is due to Niels Bohr, and has been pursued in detail by Heinz Dieter Zeh. Cited on page 149. 101 G. Lindblad, On the generators of quantum dynamical subgroups, Communications in Mathematical Physics 48, pp. 119–130, 1976. Cited on page 149. 102 Wojciech H. Zurek, Decoherence and the transition from quantum to classical, Physics Today pp. 36–44, October 1991. An easy but somewhat confusing article. His reply to the numerous letters of response in Physics Today, April 1993, pp. 13–15, and pp. 81–90, exposes his ideas in a clearer way and gives a taste of the heated discussions on this topic. Cited on copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net pages 149 and 156. 103 John Bardeen, explained this regularly in the review talks he gave at the end of his life, such as the one the author heard in Tokyo in the year 1990. Cited on page 150. 104 Collapse times have been measured for the first time by the group of Serge Har- oche in Paris. See M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J. M. Raimond & S. Haroche, Observing the progressive deco- herence of the “meter” in a quantum measurement, Physical Review Letters 77, pp. 4887– 4890, 1996. See also C. Guerlin, J. Bernu, S. Deléglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. -M. Raimond & S. Haroche, Progressive field-state collapse and quantum non-demolition photon counting, Nature 448, pp. 889–893, 2007. Cited on pages 150 and 162. 105 Later experiments confirming the numerical predictions from decoherence were published by C. Monroe, D. M. Meekhof, B. E. King & D. J. Wineland, A “Schrödinger cat” superposition state of an atom, Science 272, pp. 1131–1136, 1996, W. P. Schleich, Quantum physics: engineering decoherence, Nature 403, pp. 256–257, 2000, C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe & D. J. Wineland, Decoherence of quantum superpositions through coupling to engineered reservoirs, Nature 403, pp. 269–273, 2000. See also the summary by W. T. Strunz, G. Alber & F. Haake, Dekohärenz in offenen Quantensystemen, Physik Journal 1, pp. 47–52, November 2002. Cited on page 150. 106 L. Hackermüller, K. Hornberger, B. Brezger, A. Zeilinger & M. Arndt, Decoherence of matter waves by thermal emission of radiation, Nature 427, pp. 711–714, 2004. Cited on page 150. bibliography 257 107 K. Baumann, Quantenmechanik und Objektivierbarkeit, Zeitschrift für Naturforschung 25a, pp. 1954–1956, 1970. Cited on page 151. 108 See for example D. Styer, Physics Today p. 11, September 2000. Cited on page 153. 109 David B ohm, Quantum Theory, Prentice-Hall, 1951, pp. 614–622. Cited on page 154. 110 A. Einstein, B. Podolsky & N. Rosen, Can quantum-mechanical description of real- ity be considered complete?, Physical Review 48, pp. 696–702, 1935. Cited on page 154. 111 A. Aspect, J. Dalibard & G. Roger, Experimental tests of Bell’s inequalities using time-varying analyzers, Physical Review Letters 49, pp. 1804–1807, 1982, Cited on page 155. 112 G. C. Hergerfeldt, Causality problems for Fermi’s two-atom system, Physical Review Letters 72, pp. 596–599, 1994. Cited on page 155. 113 An experimental measurement of superpositions of left and right flowing cur- rents with 1010 electrons was J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal & S. Lloyd, Josephson persistent-current qubit, Science 285, pp. 1036–1039, 1999. In the year 2000, superpositions of 1 μA clockwise and anticlockwise have been detected; for more details, see J.R. Friedman & al., Quantum superposition Motion Mountain – The Adventure of Physics of distinct macroscopic states, Nature 406, p. 43, 2000. Cited on page 156. 114 On the superposition of magnetization in up and down directions there are numerous papers. Recent experiments on the subject of quantum tunnelling in magnetic systems are described in D. D. Awschalom, J. F. Smith, G. Grinstein, D. P. DiVicenzo & D. Loss, Macroscopic quantum tunnelling in magnetic proteins, Physical Review Letters 88, pp. 3092–3095, 1992, and in C. Paulsen & al., Macroscopic quantum tunnelling effects of Bloch walls in small ferromagnetic particles, Europhysics Letters 19, pp. 643–648, 1992. Cited on page 156. 115 For example, superpositions were observed in Josephson junctions by R. F. Voss & R. A. Webb, Macroscopic quantum tunnelling in 1 mm Nb Josephson junctions, Physical copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Review Letters 47, pp. 265–268, 1981, Cited on page 156. 116 S. Haroche, Entanglement, decoherence and the quantum-classical transition, Physics Today 51, pp. 36–42, July 1998. An experiment putting atom at two places at once, distant about 80 nm, was published by C. Monroe, C. Monroe, D. M. Meekhof, B. E. King & D. J. Wineland, A ‘Schroedinger Cat’ Superposition of an Atom, Science 272, pp. 1131–1136, 1996. Cited on page 156. 117 M. R. Andrews, C. G. Townsend, H. -J. Miesner, D. S. Durfee, D. M. Kurn & W. Ketterle, Observations of interference between two Bose condensates, Science 275, pp. 637–641, 31 January 1997. See also the www.aip.org/physnews/special.htm website. Cited on page 156. 118 A clear discussion can be found in S. Haroche & J. -M. Raimond, Quantum comput- ing: dream or nightmare?, Physics Today 49, pp. 51–52, 1996, as well as the comments in Physics Today 49, pp. 107–108, 1996. Cited on page 157. 119 The most famous reference on the wave function collapse is chapter IV of the book by Kurt Gottfried, Quantum Mechanics, Benjamin, New York, 1966. It was the favour- ite reference by Victor Weisskopf, and cited by him on every occasion he talked about the topic. Cited on page 158. 120 The prediction that quantum tunnelling could be observable when the dissipative interac- tion with the rest of the world is small enough was made by Leggett; the topic is reviewed in A. J. Leggett, S. Chahravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg & W. Zwerger, Dynamics of dissipative 2-state systems, Review of Modern Physics 59, pp. 1– 85, 1987. Cited on page 160. 258 bibliography 121 S. Kochen & E. P. Specker, The problem of hidden variables in quantum mechanics, Journal of Mathematics and Mechanics 17, pp. 59–87, 1967. Cited on page 163. 122 J. F. Clauser, M. A. Horne, A. Shimony & R. A. Holt, Proposed experiment to test local hidden-variable theories, Physical Review Letters 23, pp. 880–884, 1969. The more gen- eral and original result is found in J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, p. 195, 1964. Cited on page 164. 123 D. M. Greenberger, M. A. Horne & A. Zeilinger, Going beyond Bell’s the- orem, postprint of the 1989 paper at arxiv.org/abs/0712.0912. The first observation was D. B ouwmeester, J. -W. Pan, M. Daniell, H. Weinfurter & A. Zeilinger, Observation of three-photon Greenberger-Horne–Zeilinger entanglement, preprint at arxiv. org/abs/quant-ph/9810035. Cited on page 164. 124 Bryce de Witt & Neill Graham, eds., The Many–Worlds Interpretation of Quantum Mechanics, Princeton University Press, 1973. This interpretation talks about entities which cannot be observed, namely the many worlds, and often assumes that the wave function of the universe exists. Both habits are beliefs and in contrast with facts. Cited on page 167. Motion Mountain – The Adventure of Physics 125 ‘On the other had I think I can safely say that nobody understands quantum mechan- ics.’ From Richard P. Feynman, The Character of Physical Law, MIT Press, Cambridge, 1965, p. 129. He repeatedly made this statement, e.g. in the introduction of his otherwise ex- cellent QED – The Strange Theory of Light and Matter, Penguin Books, 1990. Cited on page 167. 126 M. Tegmark, The importance of quantum decoherence in brain processes, Physical Review D 61, pp. 4194–4206, 2000, or also arxiv.org/abs/quant-ph/9907009. Cited on page 168. 127 Connections between quantum theory and information theory can be followed in the In- ternational Journal of Quantum Information. Cited on page 169. 128 J. A. Wheeler, pp. 242–307, in Batelle Recontres: 1967 Lectures in Mathematics and Phys- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net ics, C. DeWitt & J. A. Wheeler, editors, W.A. Benjamin, 1968. For a pedagogical ex- planation, see John W. Norbury, From Newton’s laws to the Wheeler-DeWitt equation, arxiv.org/abs/physics/980604 or European Journal of Physics 19, pp. 143–150, 1998. Cited on page 170. 129 The most fascinating book on the topic is by Kurt Nassau, The Physics and Chemistry of Color – the Fifteen Causes of Color, 1983, and the excellent webexhibits.org/causesofcolour website. Cited on page 171. 130 Y. Ruiz-Morales & O. C. Mullins, Measured and Simulated Electronic Ab- sorption and Emission Spectra of Asphaltenes, Energy & Fuels 23, pp. 1169–1177, 2009. U. Bergmann, H. Groenzin, O. C. Mullins, P. Glatzel, J. Fetzer & S. P. Cramer, Carbon K-edge X-ray Raman spectroscopy supports simple, yet powerful de- scription of aromatic hydrocarbons and asphaltenes, Chemical Physics Letters 369, pp. 184– 191, 2003. Cited on page 171. 131 Two excellent reviews with numerous photographs are E. Grotewohl, The genetics and biochemistry of floral pigments, Annual Reviews of Plant Biology 57, pp. 761–780, 2006, and Y. Tanaka, N. Sasaki & A. Ohmiya, Biosynthesis of plant pigments: anthocyanins, betalains and carotenoids, The Plant Journal 54, pp. 733–749, 2008. Cited on page 179. 132 L. Pérez-Rodriguez & J. Viñuda, Carotenoid-based bill and eye coloration as honest signals of condition: an experimental test in the red-legged partridge (Alectoris rufa), Natur- wissenschaften 95, pp. 821–830, 2008, Cited on page 179. 133 R. Pello, D. Schaerer, J. Richard, J. -F. Le B orgne & J. -P. Kneib, ISAAC/VLT observations of a lensed galaxy at z=10.0, Astronomy and Astrophysics 416, p. L35, 2004. bibliography 259 Cited on page 182. 134 A pedagogical introduction is given by L. J. Curtis & D. G. Ellis, Use of the Einstein– Brillouin–Keller action quantization, American Journal of Physics 72, pp. 1521–1523, 2004. See also the introduction of A. Klein, WKB approximation for bound states by Heisenberg matrix mechanics, Journal of Mathematical Physics 19, pp. 292–297, 1978. Cited on pages 183 and 188. 135 J. Neukammer & al., Spectroscopy of Rydberg atoms at 𝑛 ∼ 500, Physical Review Letters 59, pp. 2947–2950, 1987. Cited on page 186. 136 Mark P. Silverman, And Yet It Moves: Strange Systems and Subtle Questions in Physics, Cambridge University Press 1993. A beautiful book by an expert on motion. Cited on pages 187, 194, and 195. 137 This is explained by J. D. Hey, Mystery error in Gamow’s Tompkins reappears, Physics Today pp. 88–89, May 2001. Cited on page 187. 138 The beautiful experiment was first published in A. S. Stodolna, A. Rouzée, F. Lépine, S. Cohen, F. Robicheaux, A. Gijsbertsen, J. H. Jungmann, Motion Mountain – The Adventure of Physics C. B ordas & M. J. J. Vrakking, Hydrogen atoms under magnification: direct obser- vation of the nodal structure of Stark states, Physical Review Letters 110, p. 213001, 2013. Cited on pages 185 and 186. 139 L. L. Foldy, The electromagnetic properties of Dirac particles, Physical Review 83, pp. 688– 693, 1951. L. L. Foldy, The electron–neutron interaction, Physical Review 83, pp. 693–696, 1951. L. L. Foldy, Electron–neutron interaction, Review of Modern Physics 30, pp. 471– 481, 1952. Cited on page 190. 140 H. Euler & B. Kockel, Über die Streuung von Licht an Licht nach der Diracschen The- orie, Naturwissenschaften 23, pp. 246–247, 1935, H. Euler, Über die Streuung von Licht an Licht nach der Diracschen Theorie, Annalen der Physik 26, p. 398, 1936, W. Heisenberg copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net & H. Euler, Folgerung aus der Diracschen Theorie des Electrons, Zeitschrift für Physik 98, pp. 714–722, 1936. Cited on page 193. 141 See the simple explanation by L. J. F. Hermans, Blue skies, blue seas, Europhysics News 37, p. 16, 2006, and the detailed explanation by C. L. Braun & S. N. Smirnov, Why is water blue?, Journal of Chemical Education 70, pp. 612–614, 1993. Cited on page 194. 142 The discovery is published in T. Friedmann & C. R. Hagen, Quantum Mechanical De- rivation of the Wallis Formula for π , Journal of Mathematical Physics 56, p. 112101, 2015, preprint at arxiv.org/1510.07813. See also I Chashchina & Z. K. Silagadze, On the quantum mechanical derivation of the Wallis formula for π, preprint at arxiv.org/1704.06153. Cited on page 194. 143 For the atomic case, see P. L. Gould, G. A. Ruff & D. E. Pritchard, Diffraction of atoms by light: the near resonant Kapitza–Dirac effect, Physical Review Letters 56, pp. 827– 830, 1986. Many early experimental attempts to observe the diffraction of electrons by light, in particular those performed in the 1980s, were controversial; most showed only the deflec- tion of electrons, as explained by H. Batelaan, Contemporary Physics 41, p. 369, 2000. Later on, he and his group performed the newest and most spectacular experiment, demon- strating real diffraction, including interference effects; it is described in D. L. Freimund, K. Aflatooni & H. Batelaan, Observation of the Kapitza–Dirac effect, Nature 413, pp. 142–143, 2001. Cited on page 195. 144 A single–atom laser was built in 1994 by K. An, J. J. Childs, R. R. Dasari & M. S. Feld, Microlaser: a laser with one atom in an optical resonator, Physical Review Letters 73, p. 3375, 1994. Cited on page 195. 260 bibliography 145 An introduction is given by P. Pinkse & G. Rempe, Wie fängt man ein Atom mit einem Photon?, Physikalische Blätter 56, pp. 49–51, 2000. Cited on page 195. 146 J.P. Briand & al., Production of hollow atoms by the excitation of highly charged ions in interaction with a metallic surface, Physical Review Letters 65, pp. 159–162, 1990. See also G. Marowsky & C. Rhodes, Hohle Atome und die Kompression von Licht in Plas- makanälen, Physikalische Blätter 52, pp. 991–994, Oktober 1996. Cited on page 195. 147 G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio & B. Odom, New determina- tion of the fine structure constant from the electron g value and QED, Physical Review Letters 97, p. 030802, 2006. Cited on page 196. 148 A. Sommerfeld, Zur Quantentheorie der Spektrallinien, Annalen der Physik 51, pp. 1– 94, 1916, and its continuation with the same title on pp. 125–167 in the same volume. The fine structure constant is introduced in the first paper, but Sommerfeld explains that the paper is a transcript of talks that he gave in 1915. Cited on page 196. 149 Wolf gang Pauli, Exclusion principle and quantum mechanics, Nobel lecture, 13 Decem- ber 1946, in Nobel Lectures, Physics, Volume 3, 1942–1962, Elsevier, 1964. Cited on page 197. Motion Mountain – The Adventure of Physics 150 An informative account of the world of psychokinesis and the paranormal is given by the famous professional magician James Randi, Flim-flam!, Prometheus Books, Buffalo 1987, as well as in several of his other books. See also the www.randi.org website. Cited on page 201. 151 Le Système International d’Unités, Bureau International des Poids et Mesures, Pavillon de Breteuil, Parc de Saint Cloud, 92310 Sèvres, France. All new developments concerning SI units are published in the journal Metrologia, edited by the same body. Showing the slow pace of an old institution, the BIPM launched a website only in 1998; it is now reachable at www.bipm.fr. See also the www.utc.fr/~tthomass/Themes/Unites/index.html website; this copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net includes the biographies of people who gave their names to various units. The site of its British equivalent, www.npl.co.uk/npl/reference, is much better; it provides many details as well as the English-language version of the SI unit definitions. Cited on page 205. 152 The bible in the field of time measurement is the two-volume work by J. Vanier & C. Audoin, The Quantum Physics of Atomic Frequency Standards, Adam Hilge, 1989. A popular account is Tony Jones, Splitting the Second, Institute of Physics Publishing, 2000. The site opdaf1.obspm.fr/www/lexique.html gives a glossary of terms used in the field. For precision length measurements, the tools of choice are special lasers, such as mode- locked lasers and frequency combs. There is a huge literature on these topics. Equally large is the literature on precision electric current measurements; there is a race going on for the best way to do this: counting charges or measuring magnetic forces. The issue is still open. On mass and atomic mass measurements, see Volume II, on page 71. On high-precision temperature measurements, see Volume I, on page 548. Cited on page 206. 153 The unofficial SI prefixes were first proposed in the 1990s by Jeff K. Aronson of the Uni- versity of Oxford, and might come into general usage in the future. See New Scientist 144, p. 81, 3 December 1994. Other, less serious proposals also exist. Cited on page 207. 154 For more details on electromagnetic unit systems, see the standard text by John David Jackson, Classical Electrodynamics, 3rd edition, Wiley, 1998. Cited on page 210. 155 D.J. Bird & al., Evidence for correlated changes in the spectrum and composition of cosmic rays at extremely high energies, Physical Review Letters 71, pp. 3401–3404, 1993. Cited on page 211. bibliography 261 156 P. J. Hakonen, R. T. Vuorinen & J. E. Martikainen, Nuclear antiferromagnetism in rhodium metal at positive and negative nanokelvin temperatures, Physical Review Letters 70, pp. 2818–2821, 1993. See also his article in Scientific American, January 1994. Cited on page 211. 157 A. Zeilinger, The Planck stroll, American Journal of Physics 58, p. 103, 1990. Can you Challenge 205 e find another similar example? Cited on page 211. 158 An overview of this fascinating work is given by J. H. Taylor, Pulsar timing and relativ- istic gravity, Philosophical Transactions of the Royal Society, London A 341, pp. 117–134, 1992. Cited on page 211. 159 The most precise clock built in 2004, a caesium fountain clock, had a precision of one part in 1015 . Higher precision has been predicted to be possible soon, among others by M. Takamoto, F. -L. Hong, R. Higashi & H. Katori, An optical lattice clock, Nature 435, pp. 321–324, 2005. Cited on page 211. 160 J. Bergquist, ed., Proceedings of the Fifth Symposium on Frequency Standards and Met- rology, World Scientific, 1997. Cited on page 211. Motion Mountain – The Adventure of Physics 161 See the information on D±𝑠 mesons from the particle data group at pdg.web.cern.ch/pdg. Cited on page 211. 180 162 About the long life of tantalum 180, see D. Belic & al., Photoactivation of Tam and its implications for the nucleosynthesis of nature’s rarest naturally occurring isotope, Physical Review Letters 83, pp. 5242–5245, 20 December 1999. Cited on page 212. 163 See the review by L. Ju, D. G. Blair & C. Zhao, The detection of gravitational waves, Reports on Progress in Physics 63, pp. 1317–1427, 2000. Cited on page 212. 164 See the clear and extensive paper by G. E. Stedman, Ring laser tests of fundamental physics and geophysics, Reports on Progress in Physics 60, pp. 615–688, 1997. Cited on page 212. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net 165 The various concepts are even the topic of a separate international standard, ISO 5725, with the title Accuracy and precision of measurement methods and results. A good introduction is John R. Taylor, An Introduction to Error Analysis: the Study of Uncertainties in Physical Measurements, 2nd edition, University Science Books, Sausalito, 1997. Cited on page 212. 166 P. J. Mohr, B. N. Taylor & D. B. Newell, CODATA recommended values of the fun- damental physical constants: 2010, preprint at arxiv.org/abs/1203.5425. This is the set of constants resulting from an international adjustment and recommended for international use by the Committee on Data for Science and Technology (CODATA), a body in the In- ternational Council of Scientific Unions, which brings together the International Union of Pure and Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry (IUPAC) and other organizations. The website of IUPAC is www.iupac.org. Cited on page 214. 167 Some of the stories can be found in the text by N. W. Wise, The Values of Precision, Princeton University Press, 1994. The field of high-precision measurements, from which the results on these pages stem, is a world on its own. A beautiful introduction to it is J. D. Fairbanks, B. S. Deaver, C. W. Everitt & P. F. Michaelson, eds., Near Zero: Frontiers of Physics, Freeman, 1988. Cited on page 214. 168 For details see the well-known astronomical reference, P. Kenneth Seidelmann, Ex- planatory Supplement to the Astronomical Almanac, 1992. Cited on page 219. 169 See the corresponding reference in the first volume. Cited on page 221. 170 A good reference is the Encyclopedia of Mathematics, in 10 volumes, Kluwer Academic Pub- lishers, 1988−1993. It explains most concepts used in mathematics. Spending an hour with it looking up related keywords is an efficient way to get an introduction into any part of 262 bibliography mathematics, especially into the vocabulary and the main connections. The opposite approach, to make things as complicated as possible, is taken in the de- lightful text by Carl E. Linderholm, Mathematics Made Difficult, 1971. Cited on page 223. 171 An excellent introduction into number systems in mathematics, including hyperreal (or nonstandard) numbers, quaternions, octonions, 𝑝-adic and surreal numbers, is the book by Heinz-Dieter Ebbinghaus, Hans Hermes, Friedrich Hirzebruch, Max Koecher, Klaus Mainzer, Jürgen Neukirch, Alexander Prestel & Reinhold Remmert, Zahlen, 3rd edition, Springer Verlag, 1993. It is also available in English, under the title Numbers, Springer Verlag, 1990. Cited on pages 225, 234, and 235. 172 For a book on how to use hyperreals in secondary school, see Helmut Wunderling, Analysis als Infinitesimalrechnung, Duden Paetec Schulbuchverlag, 2007. Cited on page 235. 173 A. Waser, Quaternions in Electrodynamics, 2001. The text can be downloaded from vari- ous websites. Cited on pages 227 and 232. 174 S. L. Altman, Rotations, Quaternions and Double Groups, Clarendon Press, 1986, and also Motion Mountain – The Adventure of Physics S. L. Altman, Hamilton, Rodriguez and the quaternion scandal, Mathematical Magazine 62, pp. 291–308, 1988. See also J. C. Hart, G. K. Francis & L. H. Kauffman, Visu- alzing quaternion rotation, ACM Transactions on Graphics 13, pp. 256–276, 1994. The latter can be downloaded in several places via the internet. Cited on page 230. 175 See the fine book by Louis H. Kauffman, Knots and Physics, World Scientific, 2nd edi- tion, 1994, which gives a clear and visual introduction to the mathematics of knots and their main applications to physics. Cited on page 231. 176 Gaussian integers are explored by G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, Clarendon Press, Oxford, 1979, in the sections 12.2 ‘The Rational Integers, the Gaussian Integers, and the Integers’, pp. 178–180, and 12.6 ‘Properties copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net of the Gaussian Integers’ pp. 182–183. For challenges relating to Gaussian integers, look at www.mathpuzzle.com/Gaussians.html. Cited on page 235. 177 About transfinite numbers, see the delightful paperback by Rudy Rucker, Infinity and the Mind – the Science and Philosophy of the Infinite, Bantam, 1983. Cited on page 235. 178 E. I. Butikov, The rigid pendulum – an antique but evergreen physical model, European Journal of Physics 20, pp. 429–441, 1999. D. Easton, The quantum mechanical tipping pen- cil – a caution for physics teachers, European Journal of Physics 28, pp. 1097–1104, 2007, Cited on page 242. C R E DI T S Acknowled gements Many people who have kept their gift of curiosity alive have helped to make this project come true. Most of all, Peter Rudolph and Saverio Pascazio have been – present or not – a constant reference for this project. Fernand Mayné, Ata Masafumi, Roberto Crespi, Serge Pahaut, Luca Bombelli, Herman Elswijk, Marcel Krijn, Marc de Jong, Martin van der Mark, Kim Jalink, my Motion Mountain – The Adventure of Physics parents Peter and Isabella Schiller, Mike van Wijk, Renate Georgi, Paul Tegelaar, Barbara and Edgar Augel, M. Jamil, Ron Murdock, Carol Pritchard, Richard Hoffman, Stephan Schiller, Franz Aichinger and, most of all, my wife Britta have all provided valuable advice and encouragement. Many people have helped with the project and the collection of material. Most useful was the help of Mikael Johansson, Bruno Barberi Gnecco, Lothar Beyer, the numerous improvements by Bert Sierra, the detailed suggestions by Claudio Farinati, the many improvements by Eric Shel- don, the detailed suggestions by Andrew Young, the continuous help and advice of Jonatan Kelu, the corrections of Elmar Bartel, and in particular the extensive, passionate and conscientious help of Adrian Kubala. Important material was provided by Bert Peeters, Anna Wierzbicka, William Beaty, Jim Carr, copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net John Merrit, John Baez, Frank DiFilippo, Jonathan Scott, Jon Thaler, Luca Bombelli, Douglas Singleton, George McQuarry, Tilman Hausherr, Brian Oberquell, Peer Zalm, Martin van der Mark, Vladimir Surdin, Julia Simon, Antonio Fermani, Don Page, Stephen Haley, Peter Mayr, Allan Hayes, Igor Ivanov, Doug Renselle, Wim de Muynck, Steve Carlip, Tom Bruce, Ryan Budney, Gary Ruben, Chris Hillman, Olivier Glassey, Jochen Greiner, squark, Martin Hard- castle, Mark Biggar, Pavel Kuzin, Douglas Brebner, Luciano Lombardi, Franco Bagnoli, Lu- kas Fabian Moser, Dejan Corovic, Paul Vannoni, John Haber, Saverio Pascazio, Klaus Finken- zeller, Leo Volin, Jeff Aronson, Roggie Boone, Lawrence Tuppen, Quentin David Jones, Arnaldo Uguzzoni, Frans van Nieuwpoort, Alan Mahoney, Britta Schiller, Petr Danecek, Ingo Thies, Vi- taliy Solomatin, Carl Offner, Nuno Proença, Elena Colazingari, Paula Henderson, Daniel Darre, Wolfgang Rankl, John Heumann, Joseph Kiss, Martha Weiss, Antonio González, Antonio Mar- tos, André Slabber, Ferdinand Bautista, Zoltán Gácsi, Pat Furrie, Michael Reppisch, Enrico Pasi, Thomas Köppe, Martin Rivas, Herman Beeksma, Tom Helmond, John Brandes, Vlad Tarko, Na- dia Murillo, Ciprian Dobra, Romano Perini, Harald van Lintel, Andrea Conti, François Belfort, Dirk Van de Moortel, Heinrich Neumaier, Jarosław Królikowski, John Dahlman, Fathi Namouni, Paul Townsend, Sergei Emelin, Freeman Dyson, S.R. Madhu Rao, David Parks, Jürgen Janek, Daniel Huber, Alfons Buchmann, William Purves, Pietro Redondi, Damoon Saghian, Wladi- mir Egorov, Markus Zecherle, Miles Mutka, plus a number of people who wanted to remain unnamed. The software tools were refined with extensive help on fonts and typesetting by Michael Zedler and Achim Blumensath and with the repeated and valuable support of Donald Arseneau; help came also from Ulrike Fischer, Piet van Oostrum, Gerben Wierda, Klaus Böhncke, Craig Up- right, Herbert Voss, Andrew Trevorrow, Danie Els, Heiko Oberdiek, Sebastian Rahtz, Don Story, 264 credits Vincent Darley, Johan Linde, Joseph Hertzlinger, Rick Zaccone, John Warkentin, Ulrich Diez, Uwe Siart, Will Robertson, Joseph Wright, Enrico Gregorio, Rolf Niepraschk and Alexander Grahn. The typesetting and book design is due to the professional consulting of Ulrich Dirr. The typography was much improved with the help of Johannes Küster and his Minion Math font. The design of the book and its website also owe much to the suggestions and support of my wife Britta. I also thank the lawmakers and the taxpayers in Germany, who, in contrast to most other countries in the world, allow residents to use the local university libraries. From 2007 to 2011, the electronic edition and distribution of the Motion Mountain text was generously supported by the Klaus Tschira Foundation. Film credits The hydrogen orbital image and animation of page 80 were produced with a sponsored copy of Dean Dauger’s software package Atom in a Box, available at daugerresearch.com. The coloured animations of wave functions on page 90, page 94, page 95, page 99, page 110, page 190 and Motion Mountain – The Adventure of Physics page 192 are copyright and courtesy by Bernd Thaller; they can be found on his splendid website vqm.uni-graz.at and in the CDs that come with his two beautiful books, Bernd Thaller, Visual Quantum Mechanics Springer, 2000, and Bernd Thaller, Advanced Visual Quantum Mechanics Springer, 2004. These books are the best one can read to get an intuitive understanding for wave functions and their evolution. The animation of the belt trick on page 131 is copyright and courtesy by Greg Egan; it can be found on his website www.gregegan.net/APPLETS/21/21. html. The beautiful animation of the belt trick on page 131 and the wonderful and so far unique animation of the fermion exchange on page 134 are copyright and courtesy of Antonio Martos. They can be found at vimeo.com/62228139 and vimeo.com/62143283. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Image credits The photograph of the east side of the Langtang Lirung peak in the Nepalese Himalayas, shown on the front cover, is courtesy and copyright by Kevin Hite and found on his blog thegettingthere. com. The photograph of a glow worm on page 14 is copyright and courtesy of John Tyler, and found on his beautiful website at www.johntyler.co.uk/gwfacts.htm. The photograph of a glass butterfly on page 16 is copyright and courtesy of Linda de Volder and found on her site at www.flickr.com/photos/lindadevolder. The photograph of a train window on page 33 is copy- right and courtesy of Greta Mansour and found at her website www.flickr.com/photos/wireful/. The graphics of the colour spectrum on page 41 is copyright and courtesy of Andrew Young and explained on his website mintaka.sdsu.edu/GF/explain/optics/rendering.html. The images of photographic film on page 42 are copyright and courtesy of Rich Evans. The images of pho- tomultipliers on page 42 are copyright and courtesy of Hamamatsu Photonics. The pictures of the low-intensity photon interference experiment of page 43 are copyright of the Delft Univer- sity of Technology, courtesy of Silvania Pereira, and found on the website www.optica.tn.tudelft. nl/education/photons.asp. The photograph of the Compton effect apparatus on page 46 was taken by Helene Hoffmann and is courtesy of Arne Gerdes from the University of Göttingen; it is found at the physics teaching website lp.uni-goettingen.de. The graph on page 50 is cour- tesy and copyright of Rüdiger Paschotta and found in his free and wonderful laser encyclopedia at www.rp-photonics.com. The photograph of the Mach–Zehnder interferometer on page 51 is copyright and courtesy of Félix Dieu and Gaël Osowiecki and found on their websites www. flickr.com/photos/felixdieu/sets/72157622768433934/ and www.flickr.com/photos/gaeloso/sets/ 72157623165826538/. The photograph on page page 53 is copyright of John Davis and courtesy credits 265 of . The telescope mirror interference image on page page 57 is copyright and courtesy of Mel Bartels and found on his site www.bbastrodesigns.com. The speckle pattern image is copyright and courtesy of Epzcaw and found on Wikimedia Commons. On page page 58, the double slit in- terference patterns are copyright and courtesy of Dietrich Zawischa and found on his website on beauty and science at www.itp.uni-hannover.de/~zawischa. The interference figure of Gaussian beams is copyright and courtesy of Rüdiger Paschotta and found on his free laser encyclope- dia at www.rp-photonics.com. The blue sky photograph on page 69 is courtesy and copyright of Giorgio di Iorio, and found on his website www.flickr.com/photos/gioischia/. The images about the wire contact experiment on page 69 is courtesy and copyright of José Costa-Krämer and AAPT. The famous photograph of electron diffraction on page 76 is copyright and courtesy of Claus Jönsson. The almost equally famous image that shows the build-up of electron diffraction on page 77 is courtesy and copyright of Tonomura Akira/Hitachi: it is found on the www.hqrd. hitachi.co.jp/em/doubleslit.cfm website. The hydrogen graph on page 85 is courtesy and copy- right of Peter Eyland. The photographs of the Aharonov–Bohm effect on page 99 are copyright and courtesy of Doru Cuturela. The images of DNA molecules on page 101 are copyright and courtesy by Hans-Werner Fink and used with permission of Wiley VCH. The experiment pictures Motion Mountain – The Adventure of Physics of the bunching and antibunching of 3 He and 4 He on page 119 are from the website atomoptic. iota.u-psud.fr/research/helium/helium.html and courtesy and copyright of Denis Boiron and Jerome Chatin. The molten metal photograph on page 172 is courtesy and copyright of Graela and found at flickr.com/photos/alaig. The sparkler photograph on page 172 is courtesy and copyright of Sarah Domingos and found at her flickr.com website. The reactor core photograph on page 172 is courtesy NASA and found on the grin.hq.nasa.gov website. The discharge lamp photographs on page 172 are courtesy and copyright of Pslawinski and found at www.wikimedia.org. The au- rora photograph on page 172 is courtesy and copyright of Jan Curtis and found at his climate.gi. alaska.edu/Curtis/curtis.html website. The coloured flames photograph on page 172 is courtesy and copyright of Philip Evans and found at his community.webshots.com/user/hydrogen01 web- copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net site. The iceberg photograph on page 173 is courtesy and copyright of Marc Shandro and found at his flickr.com/photos/mshandro website. The malachite photograph on page 173 is copyright and courtesy of Stephan Wolfsried and found on the www.mindat.org website. The shadow mask photograph on page 173 is courtesy and copyright of Planemad and found at the www.wikimedia. org website. The mineral photographs on page 173 and later are copyright and courtesy of Rob Lavinsky at irocks.com, and taken from his large and beautiful collection there and at www. mindat.org/photo-49529.html. The narcissus photograph on page 174 is courtesy and copyright of Thomas Lüthi and found at his website www.tiptom.ch/album/blumen/. The photograph with a finger with blood on page 174 is courtesy and copyright of Ian Humes and found at his website www.flickr.com/photos/ianhumes. The berries photograph on page 174 is courtesy and copyright of Nathan Wall and found at his website www.flickr.com/photos/ozboi-z. The photograph of a red-haired woman on page 174 is by dusdin and courtesy of Wikimedia. The rare photograph of a living angler fish on page 174 is courtesy and copyright of Steve Haddock and found at his website www.lifesci.uscb.edu/~biolum/. The magnetite photograph on page 175 is copyright and courtesy of Stephan Wolfsried and found on the www.mindat.org website. The desert photo- graph on page 175 is copyright of Evelien Willemsen, courtesy Raf Verbeelen and found at www. flickr.com/photos/verbeelen. The tenor saxophone photograph on page 175 is courtesy and copy- right of Selmer at www.selmer.fr. The photograph of zinc oxide on page 175 is by Walkerma and courtesy of Wikimedia. The fluorescing quantum dot photograph on page 175 is courtesy and copyright of Andrey Rogach, Center for Nanoscience, München. The zirconia photograph on page 176 is courtesy and copyright of Gregory Phillips and found at the commons.wikimedia. org website. The Tokyo sunset on page 176 is courtesy and copyright of Altus Plunkett and found at his www.flickr.com/photos/altus website. The blue quartz photograph on page 176 is courtesy 266 credits and copyright 2008 of David K. Lynch and found at his www.thulescientific.com website. The snowman photograph on page 177 is courtesy and copyright of Andreas Kostner and found at his www.flickr.com/photos/bytesinmotion website. The endangered blue poison frog photograph on page 177 is courtesy and copyright of Lee Hancock and found at the www.treewalkers.org website. The ruby glass photograph on page 177 is courtesy and copyright of the Murano Glass Shop and is found at their murano-glass-shop.it website. The photograph of a ring laser with second har- monic generation on page 177 is courtesy and copyright of Jeff Sherman and found at his flickr. com/photos/fatllama website. The abalone photograph on page 177 is courtesy and copyright of Anne Elliot and found at her flickr.com/photos/annkelliot website. The photograph of polariza- tion colours on page 177 is copyright of Nevit Dilmen and courtesy of Wikimedia. The mallard duck photograph on page 178 is courtesy and copyright of Simon Griffith and found at his www. pbase.com/simon2005 website. The opal photograph on page 178 is courtesy and copyright of Opalsnopals and found at his www.flickr.com website. The aeroplane condensation photograph on page 178 is courtesy and copyright of Franz Kerschbaum and found at the epod.usra.edu web- site. The CD photograph on page 178 is courtesy and copyright of Alfons Reichert and found at his www.chemiephysikskripte.de/artikel/cd.htm website. The liquid crystal pattern on page 178 Motion Mountain – The Adventure of Physics is courtesy and copyright of Ingo Dierking and Wiley/VCH; it is found in his wonderful book Ingo Dierking, Textures of Liquid Crystals, Wiley-VCH, 2003. See also his website reynolds. ph.man.ac.uk/people/staff/dierking/gallery. The measured colour spectrum on page 180 is copy- right and courtesy of Nigel Sharp, NOAO, FTS, NSO, KPNO, AURA and NSF. The photograph of a hydrogen discharge on page 181 is copyright and courtesy of Jürgen Bauer and found at the beautiful website www.smart-elements.com. The illustrations of hydrogen orbitals on page 187 are courtesy of Wikimedia. The images of the nodal atomic structures on page 185 are courtesy of Aneta Stodolna and copyright and courtesy of the American Physical Society; they are found at journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.213001. The graphs of the squeezed light states on page 241 are courtesy of G. Breitenbach and S. Schiller and copyright of Macmillan. copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net The historical portraits of physicists in the text are in the public domain, except where men- tioned. The photograph on the back cover, of a basilisk running over water, is courtesy and copy- right by the Belgian group TERRA vzw and found on their website www.terravzw.org. All drawings are copyright by Christoph Schiller. If you suspect that your copyright is not correctly given or obtained, this has not been done on purpose; please contact me in this case. NA M E I N DE X A A Bardeen, John 256 Bohr, Niels 17, 37, 65, 78, 79, AAPT AAPT 75, 265 Barnett, S.M. 251 143, 144, 166, 182, 256 Aedini, J. 250 Bartel, Elmar 263 life 17 Aflatooni, K. 259 Bartels, Mel 57, 265 Boiron, D. 254 Motion Mountain – The Adventure of Physics Aharonov, Y. 252 Batelaan, H. 259 Boiron, Denis 119, 265 Aichinger, Franz 263 Bauer, Jürgen 181, 266 Bombelli, Luca 263 Alber, G. 256 Baumann, K. 257 Boone, Roggie 263 Allison, S.W. 252 Bautista, Ferdinand 263 Bordas, C. 259 Altman, S.L. 262 Baylor, D.A. 249 Borgne, J.-F. Le 258 An, K. 259 Beaty, William 263 Born, Max 25, 87, 89, 248, 252 Anderson, Carl 192 Beeksma, Herman 263 life 22 Anderson, M.H. 250 Belfort, François 263 Bose, S.N. 253 Andrews, M.R. 250, 257 Belic, D. 261 Bose, Satyenra Nath Anton, A. 250 Belinfante, F.J. 255 life 118 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net APS 185 Bell, J. 163 Bouwmeester, D. 254, 258 Aquinas, Thomas 136 Bell, John 258 Bradley, C.C. 250 Arndt, M. 251, 256 life 164 Brahmagupta 223 Aronson, Jeff K. 260, 263 Belsley, M. 254 Brandes, John 263 Arseneau, Donald 263 Bergmann, U. 258 Braun, C.L. 259 Aspect, A. 254, 257 Bergquist, J. 261 Brebner, Douglas 263 Aspect, Alain 155 Bernu, J. 256 Breitenbach, G. 241, 249, 266 Aspelmeyer, M. 250 Berroir, J.-M. 254 Brendel, J. 250 Ata Masafumi 263 Berry, M.V. 255 Brezger, B. 256 Audoin, C. 260 Berry, Michael 255 Briand, J.P. 195, 260 Augel, Barbara 263 Bessel 180 Brillouin, Léon 182 Augel, Edgar 263 Beutelspacher, Albrecht 169 Broglie, L. de 251 Awschalom, D.D. 257 Beyer, Lothar 263 Broglie, Louis de 76 Biggar, Mark 263 life 34 B Bird, D.J. 260 Bronshtein, Matvei 8 Babinet, Jacques Björk, G. 249 Brown, R. Hanbury 249 life 206 Blair, D.G. 261 Bruce, Tom 263 Bader, Richard F.W. 247 Blumensath, Achim 263 Brumberg, E.M. 40, 248 Baez, John 263 Boas, P. Van Emde 253 Brune, M. 256 Bagnoli, Franco 263 Bocquillon, E. 254 Bub, J. 248 Balibar, Françoise 248, 255 Bohm, D. 252 Buchmann, Alfons 263 Balmer, Johann 182 Bohm, David 257 Budney, Ryan 263 Barberi Gnecco, Bruno 263 Bohr, N. 247, 250, 251 Bunsen, Robert 180 268 name index Busch, Paul 248 Curtis, L.J. 259 Dyson, Freeman 263 Butikov, E.I. 262 Cuturela, Doru 99, 265 Buzek, V. 254 E Böhncke, Klaus 263 D Easton, D. 262 Daffertshofer, A. 254 Ebbinghaus, Heinz-Dieter 262 C Dahlman, John 263 Egan, Greg 131, 264 Caldeira, A.O. 256 Dalibard, J. 257 Egorov, Wladimir 263 Candela, D. 251 Danecek, Petr 263 Einstein, A. 249, 253, 257 Carlip, Steve 263 Daniell, M. 258 Einstein, Albert 30, 55, 56, 105, Carr, Jim 263 Darley, Vincent 264 118, 182, 252 Carruthers, P. 251 Darre, Daniel 263 Ekstrom, C.R. 252 B Cato, Marcus Porcius 171 Cavanna, A. 254 Dasari, R.R. 259 Dauger, Dean 80, 264 Elitzur, Avshalom 69 Elliot, Anne 177, 266 Cayley, Arthur 233, 234 Davis, John 53, 264 Ellis, D.G. 259 Busch Center for Nanoscience, Davis, K.B. 250 Els, Danie 263 München 265 Deaver, B.S. 261 Elswijk, Herman B. 263 Motion Mountain – The Adventure of Physics Chahravarty, S. 257 Degen, Carl Ferdinand 234 Emelin, Sergei 263 Chang, H. 254 Degiovanni, P. 254 Engels, F. 248 Chang, I. 249 Dehmelt, H. 253, 254 Engels, Friedrich 39, 74 Chapman, M.S. 252 Dehmelt, Hans 254 Englert, Berthold-Georg 248 Chashchina, O.I 259 Delft University of Epicurus 40, 44 Chatin, Jerome 119, 265 Technology 43, 264 Epzcaw 57, 265 Childs, J.J. 259 Deléglise, S. 256 Erdős, Paul Chu, Steven 195 DeWitt, C. 258 life 223 Cicero, Marcus Tullius 72 Dicke, Robert H. 251 Erschow, A. 255 Cirone, M.A. 253 Dieks, D. 254 Euler, H. 259 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Clauser, J.F. 258 Dieks, Dennis 122 Euler, Hans 193 Clifton, R. 248 Dierking, Ingo 178, 266 Euler, Leonhard 229 Cohen, S. 259 Dieu, Félix 51, 264 Evans, Philip 172, 265 Cohen-Tannoudji, C. 248 Diez, Ulrich 264 Evans, Rich 42, 264 Cohen-Tannoudji, Claude 195 DiFilippo, Frank 263 Everitt, C.W. 261 Colazingari, Elena 263 Dilmen, Nevit 177, 266 Eyland, Peter 85, 265 Colella, R. 252 Diophantus of Alexandria 226 Collins, D. 254 Dirac 195 F Compton, A.H. 249 Dirac, P.A.M. 250 Facchi, P. 251 Compton, Arthur 45 Dirac, Paul 59, 66, 189 Fairbanks, J.D. 261 Conti, Andrea 263 life 189 Faragó, B. 252 Corovic, Dejan 263 Dirr, Ulrich 264 Farinati, Claudio 263 Correia, A. 251 Diu, B. 248 Feld, M.S. 259 Costa-Krämer, J.L. 250, 251 DiVicenzo, D.P. 257 Fermani, Antonio 263 Costa-Krämer, José 74, 75, 265 Dobra, Ciprian 263 Fermi, Enrico Costella, J.P. 253 Domingos, Sarah 172, 265 life 118 Courtial, J. 251 Dorsey, A.T. 257 Fetzer, J. 258 Cramer, S.P. 258 Dreyer, J. 256 Feynman, R. P. 255 Crane, H.R. 253 Druten, N.J. van 250 Feynman, Richard 167 Crespi, Roberto 263 Duck, I. 255 Feynman, Richard P. 250, 258 Crommle, M.F. 242 Duran, E. 255 Feynman, Cser, L. 252 Durfee, D.S. 250, 257 Richard (‘Dick’) Phillips Curtis, Jan 172, 265 dusdin 174, 265 life 60 name index 269 Fink, Hans-Werner 101, 252, Glauber, R.J. 250 Hannout, M. 252 265 Glauber, Roy 63 Hardcastle, Martin 263 Finkenzeller, Klaus 263 Gleyzes, S. 256 Hardy, G.H. 262 Firestone, R.B. 254 González, Antonio 263 Haroche, S. 256, 257 Fischbach, E. 249 Gottfried, Kurt 257 Haroche, Serge 150, 256 Fischer, Ulrike 263 Goudsmit, S. 252 Hart, J.C. 262 Fisher, M.P.A. 257 Goudsmit, Samuel 104 Hartung, R.W. 255 Foldy, L.L. 106, 252, 259 Gould, P.L. 259 Hausherr, Tilman 263 Foley, E.L. 251 Gould, R. 255 Hayes, Allan 263 Fonseca, E.J.S. 249 Graela 172, 265 Heaviside 232 Francis, G.K. 262 Graham, Neill 258 Hegerfeldt, Gerhard 155 F Franke-Arnold, S. 251 Fraunhofer, Joseph Grahn, Alexander 264 Graves, John 234 Heisenberg, W. 259 Heisenberg, Werner 24, 25, life 180 Gray, C.G. 252 78, 193 Fink Freimund, D.L. 259 Greenberg, O.W. 254, 255 life 25 Freulon, V. 254 Greenberg, Oscar 142 Helmond, Tom 263 Motion Mountain – The Adventure of Physics Friedman, J.R. 257 Greenberger, D.M. 255, 258 Henderson, Paula 263 Friedmann, T. 259 Greenbergest, T.V. 255 Hergerfeldt, G.C. 257 Fumagalli, Giuseppe 247 Gregorio, Enrico 264 Hermann, Armin 248 Furrie, Pat 263 Greiner, Jochen 263 Hermans, L.J.F. 259 Fève, G. 254 Griffith, Simon 178, 266 Hermes, Hans 262 Grinstein, G. 257 Herneck, Friedrich 250 G Grit, C.O. 242 Hertz 232 Gabrielse, G. 254, 260 Groenzin, H. 258 Hertz, Heinrich 54, 189 Galilei, Galileo 21, 24 Grotewohl, E. 258 Hertzlinger, Joseph 264 Galle, Johann Gottfried 180 Guerlin, C. 256 Hess, Victor 92 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Garcia, N. 250, 251 Gácsi, Zoltán 263 Heumann, John 263 García-Mochales, P. 250, 251 Hewitt, Susan 248 Gardner, Martin 253 H Hey, J.D. 259 Garg, A. 257 Haake, F. 256 Higashi, R. 261 Gasiorowicz, Stephen 251 Haas, Arthur Erich 21, 186 Hilbert, David 223 Gasparoni, S. 250 Haber, John 263 Hilborn, R. 255 Gauß, Carl-Friedrich 235 Hackermüller, L. 251 Hilgevoord, Jan 248 Gelbaum, Bernard R. 238 Hackermüller, L. 256 Hillery, M. 254 Gell-Mann, M. 255 Haddock, Steve 174, 265 Hillman, Chris 263 Gell-Mann, Murray 143, 167 Hagen, C. R. 259 Hirzebruch, Friedrich 262 Georgi, Renate 263 Hagley, E. 256 Hitachi 77, 265 Gerdes, Arne 264 Hajnal, A. 253 Hite, Kevin 264 Gerlach, W. 251 Hakonen, P.J. 260 Hitler, Adolf 17 Gerlach, Walther Haley, Stephen 263 Hoffman, Richard 263 life 83 Halvorson, H. 248 Hoffmann, Helene 46, 264 Gibbs, Josiah Willard Hamamatsu Photonics 42 Hogervorst, W. 254 life 114 Hamilton 229 Holt, R.A. 258 Gijsbertsen, A. 259 Hamilton, William Rowan Hong, C.K. 253 Gillies, G.T. 252 life 227 Hong, F.-L. 261 Gisin, N. 250, 254 Hammond, T.D. 252 Hoppeler, R. 254 Glanz, J. 249 Hanbury Brown, Robert 52 Hornberger, K. 256 Glassey, Olivier 263 Hancock, Lee 177, 266 Horne, M.A. 258 Glatzel, P. 258 Hanneke, D. 260 Howell, J.C. 254 270 name index Hoyt, S. 252 Keller, C. 251 Linderholm, Carl E. 262 Huber, Daniel 263 Keller, Joseph 182 Lintel, Harald van 263 Huff, D. 255 Kells, W. 254 Lloyd, S. 257 Hulet, R.G. 250 Kelu, Jonatan 263 Lockyer, Joseph 181 Humes, Ian 174, 265 Kerschbaum, Franz 178, 266 Lombardi, Luciano 263 Hurwitz, Adolf 234 Ketterle, W. 250, 257 Loss, D. 257 Hänsch, Theodor 194 Kidd, R. 250 Loudon, Rodney 248 Kielpinski, D. 256 Louisell, W.H. 251 I King, B. E. 257 Lovász, P. 253 Icke, Vincent 248 King, B.E. 256 Lui, A.T.Y. 249 Iorio, Giorgio di 69, 265 Kinoshita, T. 260 Lundeen, J.S. 249 H Itano, W.M. 256 Ivanov, Igor 263 Kirchhoff, Gustav 180 Kiss, Joseph 263 Lynch, David 176 Lépine, F. 259 Klauder, J.R. 250 Lévy-Leblond, Jean-Marc Hoy t J Klaus Tschira Foundation 264 248, 255 Jackson, John David 260 Klein, A. 259 Lüthi, Thomas 174, 265 Motion Mountain – The Adventure of Physics Jacobson, J. 249 Klein, Oskar 192 Jalink, Kim 263 Kloor, H. 249 M Jamil, M. 263 Kneib, J.-P. 258 Maali, A. 256 Jammer, Max 247 Kochen, S. 163, 258 Macmillan 241, 266 Janek, Jürgen 263 Kockel, B. 259 Magyar G. 67 Janssen, Jules 181 Koecher, Max 262 Magyar, G. 250 Jeltes, T. 254 Kostner, Andreas 177, 266 Mahoney, Alan 263 Jin, Y. 254 Krachmalnicoff, V. 254 Mainzer, Klaus 262 Johansson, Mikael 263 Krexner, G. 252 Maitre, X. 256 Johnson, Samuel 247 Krijn, Marcel 263 Malik, J. 248 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Jones, Quentin David 263 Kronig, Ralph 104 Mandel, L. 67, 250, 253 Jones, Tony 260 Kryowonos, A. 252 Mansour, Greta 33, 264 Jong, Marc de 263 Królikowski, Jarosław 263 Mariano, A. 251 Joos, E. 256 Kubala, Adrian 263 Mark, Martin van der 263 Joos, Erich 256 Kuhr, S. 256 Marowsky, G. 260 Jordan, Pascual 24, 25 Kurn, D.M. 250, 257 Marqués, M. 251 Joseph Bohm, David Kuzin, Pavel 263 Martikainen, J.E. 261 life 154 Kuščer, I. 252 Martini, K.M. 251 Joyce, James 152 Küster, Johannes 264 Martos, Antonio 131, 134, 263, Ju, L. 261 264 Jungmann, J.H. 259 L Massar, S. 254 Jönsson, C. 251 Lagrange, Joseph 229 Mattheck, Claus 248 Jönsson, Claus 76, 77, 265 Laloë, F. 248 Maxwell 232 Lamas-Linares, A. 254 Mayné, Fernand 263 K Langel, R.A. 249 Mayr, Peter 263 K. Lynch, David 266 Lavinsky, Rob 173, 176, 265 McKellar, B.H.J. 253 Köppe, Thomas 263 Leach, J. 251 McNamara, J.M. 254 Kan, A.H.G. Rinnooy 253 Leggett, A.J. 256, 257 McQuarry, George 263 Kapitza 195 Lenstra, J.K. 253 Meekhof, D. M. 257 Karl, G. 252 Leonardo da Vinci 81 Meekhof, D.M. 256 Katori, H. 261 Levitov, L. 257 Mensky, M.B. 247 Kauffman, L.H. 262 Lindblad, G. 256 Merrit, John 263 Kauffman, Louis H. 262 Linde, Johan 264 Metikas, G. 253 name index 271 Mewes, M.-O. 250 O Philips, William 195 Meyenn, K. von 252 Oberdiek, Heiko 263 Phillips, Gregory 176, 265 Meyer, J.C. 242 Oberquell, Brian 263 Photonics, Hamamatsu 264 Michaelson, P.F. 261 Odom, B. 260 Pinkse, P. 260 Miesner, H.-J. 257 Offner, Carl 263 Pittman, T.B. 253 Migdall, A. 253 Ohanian, Hans C. 255 Planck, Erwin 248 Milonni, P.W. 249 Ohmiya, A. 258 Planck, Max 20, 47, 55, 105, Misner, C.W. 255 Olmsted, John M.H. 238 247 Mitchell, M.W. 249 Oostrum, Piet van 263 life 17 Mlynek, J. 249 Opalsnopals 178, 266 Planemad 173, 265 Mohr, P.J. 261 Orlando, T.P. 257 Plastino, A. 254 M Monken, C.H. 249 Monroe, C. 256, 257 Orlitzky, A. 253 Osowiecki, Gaël 51, 264 Plastino, A.R. 254 Plaçais, B. 254 Mooij, J.E. 257 Ou, Z.Y. 253 Plunkett, Altus 176, 265 Mewes Moortel, Dirk Van de 263 Overhauser, A.W. 252 Podolsky, B. 257 Moser, Lukas Fabian 263 Poincaré, Henri 55, 249 Motion Mountain – The Adventure of Physics Mullins, O.C. 258 P Popescu, S. 254 Murdock, Ron 263 Pádua, de 55 Prentis, J.J. 249 Murillo, Nadia 263 Padgett, M. 251 Prestel, Alexander 262 Mutka, Miles 263 Page, Don 263 Prewitt, O. 255 Muynck, Wim de 263 Pahaut, Serge 85, 263 Pritchard, Carol 263 Myatt, C.J. 256 Pan, J.-W. 249, 258 Pritchard, D.E. 252, 259 Papini, G. 251 Pritchard, David 85 N Parks, David 263 Proença, Nuno 263 Nagaoka Hantaro 79 Pascal, Blaise Pslawinski 172, 265 Nagaoka, H. 251 life 46 Purves, William 263 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Nairz, O. 251 Pascazio, S. 251 Pádua, S. de 249 Namouni, Fathi 263 Pascazio, Saverio 263 Pérez-Rodriguez, L. 258 NASA 172, 265 Paschotta, Rüdiger 50, 58, Nassau, Kurt 258 264, 265 R Neukammer, J. 259 Pasi, Enrico 263 Rahtz, Sebastian 263 Neukirch, Jürgen 262 Paul, H. 250 Raimond, J.-M. 256, 257 Neumaier, Heinrich 263 Pauli, W. 255 Raimond, J.M. 256 Neumann, John von 144 Pauli, Wolfgang 25, 61, 133, Ramberg, E. 122, 254 life 163 196, 260 Randi, James 260 Neumann, János life 105 Rankl, Wolfgang 263 life 163 Paulsen, C. 257 Redondi, Pietro 263 Newell, D.B. 261 Payne, Cecilia Reichert, Alfons 178, 266 Newton 70 life 181 Reichl, Linda 256 Newton, T.D. 106, 252 Peeters, Bert 263 Remmert, Reinhold 262 Nienhuis, G. 252 Pello, R. 258 Rempe, G. 260 Niepraschk, Rolf 264 Penrose, R. 255 Renselle, Doug 263 Nieto, M.M. 251 Peredo, M. 249 Reppisch, Michael 263 Nieuwpoort, Frans van 263 Pereira, Silvania 264 Rhodes, C. 260 Nio, M. 260 Peres, Asher 248 Richard, J. 258 Norbury, John W. 258 Perini, Romano 263 Rieke, F. 248, 249 Novikov, V.A. 252 Perrin, A. 254 Rindler, W. 255 Perrin, J. 251 Rivas, Martin 263 Perrin, Jean 79 Robbins, J.M. 255 272 name index Robertson, Will 264 Selmer 175, 265 Takamoto, M. 261 Robicheaux, F. 259 Serena, P.A. 250, 251 Tanaka, Y. 258 Rogach, Andrey 175, 265 Sergienko, A.V. 253 Tarko, Vlad 263 Roger, G. 257 Shandro, Marc 173, 265 Taylor, B.N. 261 Roos, Hans 248 Sharkov, I. 252 Taylor, J.H. 261 Rosen, N. 257 Sharp, Nigel 180, 266 Taylor, John R. 261 Rosenfeld 65 Shaw, George Bernard 72 Tegelaar, Paul 263 Rosenfeld, L. 250 Sheldon, Eric 263 Tegmark, M. 256, 258 Rouzée, A. 259 Shepp, L. 253 Tetrode, Hugo 114 Ruben, Gary 263 Sherman, Jeff 177, 266 Thaler, Jon 263 Rubin, M.H. 253 Shih, Y.H. 253 Thaller, Bernd 90, 94, 95, 99, R Rucker, Rudy 262 Rudolph, Peter 263 Shimony, A. 258 Siart, Uwe 264 110, 190, 192, 264 Thies, Ingo 263 Ruff, G.A. 259 Sierra, Bert 263 Thomas, L. 252 Robertson Ruiz-Morales, Y. 258 Silagadze, Z.K. 259 Thomas, Llewellyn 105 Rydberg, Johannes 182 Silverman, M.P. 252 Thorne, K.S. 255 Motion Mountain – The Adventure of Physics Silverman, Mark 194 Tian, L. 257 S Silverman, Mark P. 259 Tiberius 199 S.R. Madhu Rao 263 Simon, C. 254 Tino, G. 255 Sabbata, V. de 252 Simon, Julia 263 Tittel, Wolfgang 250 Sackett, C.A. 250, 256 Singleton, Douglas 263 Tollett, J.J. 250 Sackur, Otto 114 Sivaram, C. 252 Tonomura Akira 77, 265 Sagan, Hans 238 Slabber, André 263 Townsend, C.G. 257 Saghian, Damoon 263 Smirnov, S.N. 259 Townsend, Paul 263 Salam, Abdus 248 Smith, J.F. 257 Trevorrow, Andrew 263 Sasaki, N. 258 Snow, G.A. 122, 254 Tschira, Klaus 264 copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net Sayrin, C. 256 Solomatin, Vitaliy 263 Tuominen, M.T. 251 Schaerer, D. 258 Sommerfeld, A. 260 Tuppen, Lawrence 263 Schellekens, M. 254 Sommerfeld, Arnold 186, 188, Turchette, Q.A. 256 Schiller, Britta 263, 264 196 Twamley, J. 255 Schiller, Christoph 266 Specker, E.P. 163, 258 Twiss, R.Q. 249 Schiller, Friedrich Stedman, G.E. 261 Twiss, Richard 52 life 40 Steinberg, A.M. 249 Tyler, John 16, 264 Schiller, Isabella 263 Stern, O. 251 Török, Gy. 252 Schiller, Peter 263 Stern, Otto Schiller, S. 249, 266 life 83 U Schiller, Stephan 263 Stodolna, A.S. 259 Uguzzoni, Arnaldo 263 Schleich, W.P. 253, 256 Stodolna, Aneta 185, 186, 266 Uhlenbeck, G.E. 252 Schmiedmayer, J. 252 Story, Don 263 Uhlenbeck, George 104 Schrödinger, E. 252 Strekalov, D.V. 253 Upright, Craig 263 Schrödinger, Erwin 36, 182 Strunz, W.T. 256 Ursin, R. 250 life 92 Styer, D. 257 Schubert, Max 248 Subitzky, Edward 248 V Schucking, E. 252 Sudarshan, E. C. G. 255 Vaidman, Lev 69 Schwenk, Jörg 169 Sudarshan, E.C.G. 250 Vanier, J. 260 Schwinger, Julian 102, 103, 248 Surdin, Vladimir 263 Vannoni, Paul 263 Schönenberger, C. 254 Vardi, Ilan 253 Scott, Jonathan 263 T Vassen, W. 254 Seidelmann, P. Kenneth 261 Tacitus 199 Vavilov, S.I. 40, 248 name index 273 Verbeelen, Raf 265 Wheeler, John 239 Wouthuysen, S.A. 106, 252 Viana Gomes, J. 254 Wheeler, John A. 255 Wright, E.M. 262 Vico, Giambattista Widom, A. 252 Wright, Joseph 264 life 165 Wierda, Gerben 263 Wunderlich, C. 256 Viñuda, J. 258 Wierzbicka, Anna 263 Wunderling, Helmut 262 Volder, Linda de 16, 264 Wiesner, S.J. 254 Volin, Leo 263 Wiesner, Stephen 122 Y Vos-Andreae, J. 251 Wigner, E. 254 Yamamoto, Y. 249 Voss, Herbert 263 Wigner, E.P. 106, 252 Yao, E. 251 Voss, R.F. 257 Wigner, Eugene Yazdani, A. 253 Vrakking, M.J.J. 259 life 125 Young, Andrew 41, 263, 264 V Vuorinen, R.T. 261 Wigner, Eugene P. 248 Wijk, Mike van 263 Z W Wikimedia 187, 265, 266 Zaccone, Rick 264 Verbeelen Wal, C.H. van der 257 Wiley VCH 101, 265 Zalm, Peer 263 Walkerma 175, 265 Wiley/VCH 266 Zawischa, Dietrich 58, 265 Motion Mountain – The Adventure of Physics Wall, Nathan 174, 265 Willemsen, Evelien 175, 265 Zbinden, H. 250 Walther, P. 249 Wineland, D. J. 257 Zecherle, Markus 263 Warkentin, John 264 Wineland, D.J. 256 Zedler, Michael 263 Waser, A. 262 Wise, N.W. 261 Zeh, H.D. 256 Webb, R.A. 257 Witt, Bryce de 258 Zeh, Heinz Dieter 144, 256 Weber, Gerhard 248 Wittke, James P. 251 Zeilinger, A. 250, 251, 256, 258, Wehinger,