Quantum Harmonic Oscillator in 1-D

sympy.physics.qho_1d.E_n(n, omega)

Returns the Energy of the One-dimensional harmonic oscillator.

Parameters:

``n`` :

The “nodal” quantum number.

``omega`` :

The harmonic oscillator angular frequency.

Notes

The unit of the returned value matches the unit of hw, since the energy is calculated as:

E_n = hbar * omega*(n + 1/2)

Examples

>>> from sympy.physics.qho_1d import E_n
>>> from sympy.abc import x, omega
>>> E_n(x, omega)
hbar*omega*(x + 1/2)

sympy.physics.qho_1d.coherent_state(n, alpha)

Returns <n|alpha> for the coherent states of 1D harmonic oscillator. See https://en.wikipedia.org/wiki/Coherent_states

Parameters:

``n`` :

The “nodal” quantum number.

``alpha`` :

The eigen value of annihilation operator.

sympy.physics.qho_1d.psi_n(n, x, m, omega)

Returns the wavefunction psi_{n} for the One-dimensional harmonic oscillator.

Parameters:

``n`` :

the “nodal” quantum number. Corresponds to the number of nodes in the wavefunction. n >= 0

``x`` :

x coordinate.

``m`` :

Mass of the particle.

``omega`` :

Angular frequency of the oscillator.

Examples

>>> from sympy.physics.qho_1d import psi_n
>>> from sympy.abc import m, x, omega
>>> psi_n(0, x, m, omega)
(m*omega)**(1/4)*exp(-m*omega*x**2/(2*hbar))/(hbar**(1/4)*pi**(1/4))