Operator¶
Quantum mechanical operators.
TODO:
Fix early 0 in apply_operators.
Debug and test apply_operators.
Get cse working with classes in this file.
Doctests and documentation of special methods for InnerProduct, Commutator, AntiCommutator, represent, apply_operators.
- class sympy.physics.quantum.operator.DifferentialOperator(*args, **kwargs)[source]¶
An operator for representing the differential operator, i.e. d/dx
It is initialized by passing two arguments. The first is an arbitrary expression that involves a function, such as
Derivative(f(x), x)
. The second is the function (e.g.f(x)
) which we are to replace with theWavefunction
that thisDifferentialOperator
is applied to.- Parameters
expr : Expr
The arbitrary expression which the appropriate Wavefunction is to be substituted into
func : Expr
A function (e.g. f(x)) which is to be replaced with the appropriate Wavefunction when this DifferentialOperator is applied
Examples
You can define a completely arbitrary expression and specify where the Wavefunction is to be substituted
>>> from sympy import Derivative, Function, Symbol >>> from sympy.physics.quantum.operator import DifferentialOperator >>> from sympy.physics.quantum.state import Wavefunction >>> from sympy.physics.quantum.qapply import qapply >>> f = Function('f') >>> x = Symbol('x') >>> d = DifferentialOperator(1/x*Derivative(f(x), x), f(x)) >>> w = Wavefunction(x**2, x) >>> d.function f(x) >>> d.variables (x,) >>> qapply(d*w) Wavefunction(2, x)
- property expr¶
Returns the arbitrary expression which is to have the Wavefunction substituted into it
Examples
>>> from sympy.physics.quantum.operator import DifferentialOperator >>> from sympy import Function, Symbol, Derivative >>> x = Symbol('x') >>> f = Function('f') >>> d = DifferentialOperator(Derivative(f(x), x), f(x)) >>> d.expr Derivative(f(x), x) >>> y = Symbol('y') >>> d = DifferentialOperator(Derivative(f(x, y), x) + ... Derivative(f(x, y), y), f(x, y)) >>> d.expr Derivative(f(x, y), x) + Derivative(f(x, y), y)
- property free_symbols¶
Return the free symbols of the expression.
- property function¶
Returns the function which is to be replaced with the Wavefunction
Examples
>>> from sympy.physics.quantum.operator import DifferentialOperator >>> from sympy import Function, Symbol, Derivative >>> x = Symbol('x') >>> f = Function('f') >>> d = DifferentialOperator(Derivative(f(x), x), f(x)) >>> d.function f(x) >>> y = Symbol('y') >>> d = DifferentialOperator(Derivative(f(x, y), x) + ... Derivative(f(x, y), y), f(x, y)) >>> d.function f(x, y)
- property variables¶
Returns the variables with which the function in the specified arbitrary expression is evaluated
Examples
>>> from sympy.physics.quantum.operator import DifferentialOperator >>> from sympy import Symbol, Function, Derivative >>> x = Symbol('x') >>> f = Function('f') >>> d = DifferentialOperator(1/x*Derivative(f(x), x), f(x)) >>> d.variables (x,) >>> y = Symbol('y') >>> d = DifferentialOperator(Derivative(f(x, y), x) + ... Derivative(f(x, y), y), f(x, y)) >>> d.variables (x, y)
- class sympy.physics.quantum.operator.HermitianOperator(*args, **kwargs)[source]¶
A Hermitian operator that satisfies H == Dagger(H).
- Parameters
args : tuple
The list of numbers or parameters that uniquely specify the operator. For time-dependent operators, this will include the time.
Examples
>>> from sympy.physics.quantum import Dagger, HermitianOperator >>> H = HermitianOperator('H') >>> Dagger(H) H
- class sympy.physics.quantum.operator.IdentityOperator(*args, **kwargs)[source]¶
An identity operator I that satisfies op * I == I * op == op for any operator op.
- Parameters
N : Integer
Optional parameter that specifies the dimension of the Hilbert space of operator. This is used when generating a matrix representation.
Examples
>>> from sympy.physics.quantum import IdentityOperator >>> IdentityOperator() I
- class sympy.physics.quantum.operator.Operator(*args, **kwargs)[source]¶
Base class for non-commuting quantum operators.
An operator maps between quantum states [R635]. In quantum mechanics, observables (including, but not limited to, measured physical values) are represented as Hermitian operators [R636].
- Parameters
args : tuple
The list of numbers or parameters that uniquely specify the operator. For time-dependent operators, this will include the time.
Examples
Create an operator and examine its attributes:
>>> from sympy.physics.quantum import Operator >>> from sympy import I >>> A = Operator('A') >>> A A >>> A.hilbert_space H >>> A.label (A,) >>> A.is_commutative False
Create another operator and do some arithmetic operations:
>>> B = Operator('B') >>> C = 2*A*A + I*B >>> C 2*A**2 + I*B
Operators don’t commute:
>>> A.is_commutative False >>> B.is_commutative False >>> A*B == B*A False
Polymonials of operators respect the commutation properties:
>>> e = (A+B)**3 >>> e.expand() A*B*A + A*B**2 + A**2*B + A**3 + B*A*B + B*A**2 + B**2*A + B**3
Operator inverses are handle symbolically:
>>> A.inv() A**(-1) >>> A*A.inv() 1
References
- class sympy.physics.quantum.operator.OuterProduct(*args, **old_assumptions)[source]¶
An unevaluated outer product between a ket and bra.
This constructs an outer product between any subclass of
KetBase
andBraBase
as|a><b|
. AnOuterProduct
inherits from Operator as they act as operators in quantum expressions. For reference see [R637].- Parameters
ket : KetBase
The ket on the left side of the outer product.
bar : BraBase
The bra on the right side of the outer product.
Examples
Create a simple outer product by hand and take its dagger:
>>> from sympy.physics.quantum import Ket, Bra, OuterProduct, Dagger >>> from sympy.physics.quantum import Operator >>> k = Ket('k') >>> b = Bra('b') >>> op = OuterProduct(k, b) >>> op |k><b| >>> op.hilbert_space H >>> op.ket |k> >>> op.bra <b| >>> Dagger(op) |b><k|
In simple products of kets and bras outer products will be automatically identified and created:
>>> k*b |k><b|
But in more complex expressions, outer products are not automatically created:
>>> A = Operator('A') >>> A*k*b A*|k>*<b|
A user can force the creation of an outer product in a complex expression by using parentheses to group the ket and bra:
>>> A*(k*b) A*|k><b|
References
- property bra¶
Return the bra on the right side of the outer product.
- property ket¶
Return the ket on the left side of the outer product.
- class sympy.physics.quantum.operator.UnitaryOperator(*args, **kwargs)[source]¶
A unitary operator that satisfies U*Dagger(U) == 1.
- Parameters
args : tuple
The list of numbers or parameters that uniquely specify the operator. For time-dependent operators, this will include the time.
Examples
>>> from sympy.physics.quantum import Dagger, UnitaryOperator >>> U = UnitaryOperator('U') >>> U*Dagger(U) 1