# 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 the Wavefunction that this DifferentialOperator 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 [R629]. In quantum mechanics, observables (including, but not limited to, measured physical values) are represented as Hermitian operators [R630].

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

R629(1,2)

https://en.wikipedia.org/wiki/Operator_%28physics%29

R630(1,2)

https://en.wikipedia.org/wiki/Observable

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 and BraBase as |a><b|. An OuterProduct inherits from Operator as they act as operators in quantum expressions. For reference see [R631].

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

R631(1,2)

https://en.wikipedia.org/wiki/Outer_product

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