Represent

Logic for representing operators in state in various bases.

TODO:

• Get represent working with continuous hilbert spaces.
• Document default basis functionality.

sympy.physics.quantum.represent.represent(expr, **options)

Represent the quantum expression in the given basis.

In quantum mechanics abstract states and operators can be represented in various basis sets. Under this operation the follow transforms happen:

• Ket -> column vector or function
• Bra -> row vector of function
• Operator -> matrix or differential operator

This function is the top-level interface for this action.

This function walks the sympy expression tree looking for QExpr instances that have a _represent method. This method is then called and the object is replaced by the representation returned by this method. By default, the _represent method will dispatch to other methods that handle the representation logic for a particular basis set. The naming convention for these methods is the following:

def _represent_FooBasis(self, e, basis, **options)

This function will have the logic for representing instances of its class in the basis set having a class named FooBasis.

Parameters:

expr : Expr

The expression to represent.

basis : Operator, basis set

An object that contains the information about the basis set. If an operator is used, the basis is assumed to be the orthonormal eigenvectors of that operator. In general though, the basis argument can be any object that contains the basis set information.

options : dict

Key/value pairs of options that are passed to the underlying method that finds the representation. These options can be used to control how the representation is done. For example, this is where the size of the basis set would be set.

Returns:

e : Expr

The SymPy expression of the represented quantum expression.

Examples

Here we subclass Operator and Ket to create the z-spin operator and its spin 1/2 up eigenstate. By definining the _represent_SzOp method, the ket can be represented in the z-spin basis.

>>> from sympy.physics.quantum import Operator, represent, Ket
>>> from sympy import Matrix

>>> class SzUpKet(Ket):
...     def _represent_SzOp(self, basis, **options):
...         return Matrix([1,0])
...
>>> class SzOp(Operator):
...     pass
...
>>> sz = SzOp('Sz')
>>> up = SzUpKet('up')
>>> represent(up, basis=sz)
Matrix([
[1],
[0]])

Here we see an example of representations in a continuous basis. We see that the result of representing various combinations of cartesian position operators and kets give us continuous expressions involving DiracDelta functions.

>>> from sympy.physics.quantum.cartesian import XOp, XKet, XBra
>>> X = XOp()
>>> x = XKet()
>>> y = XBra('y')
>>> represent(X*x)
x*DiracDelta(x - x_2)
>>> represent(X*x*y)
x*DiracDelta(x - x_3)*DiracDelta(x_1 - y)

sympy.physics.quantum.represent.rep_innerproduct(expr, **options)

Returns an innerproduct like representation (e.g. <x'|x>) for the given state.

Attempts to calculate inner product with a bra from the specified basis. Should only be passed an instance of KetBase or BraBase

Parameters:

expr : KetBase or BraBase

The expression to be represented

Examples

>>> from sympy.physics.quantum.represent import rep_innerproduct
>>> from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet
>>> rep_innerproduct(XKet())
DiracDelta(x - x_1)
>>> rep_innerproduct(XKet(), basis=PxOp())
sqrt(2)*exp(-I*px_1*x/hbar)/(2*sqrt(hbar)*sqrt(pi))
>>> rep_innerproduct(PxKet(), basis=XOp())
sqrt(2)*exp(I*px*x_1/hbar)/(2*sqrt(hbar)*sqrt(pi))

sympy.physics.quantum.represent.rep_expectation(expr, **options)

Returns an <x'|A|x> type representation for the given operator.

Parameters:

expr : Operator

Operator to be represented in the specified basis

Examples

>>> from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet
>>> from sympy.physics.quantum.represent import rep_expectation
>>> rep_expectation(XOp())
x_1*DiracDelta(x_1 - x_2)
>>> rep_expectation(XOp(), basis=PxOp())
<px_2|*X*|px_1>
>>> rep_expectation(XOp(), basis=PxKet())
<px_2|*X*|px_1>

sympy.physics.quantum.represent.integrate_result(orig_expr, result, **options)

Returns the result of integrating over any unities (|x><x|) in the given expression. Intended for integrating over the result of representations in continuous bases.

This function integrates over any unities that may have been inserted into the quantum expression and returns the result. It uses the interval of the Hilbert space of the basis state passed to it in order to figure out the limits of integration. The unities option must be specified for this to work.

Note: This is mostly used internally by represent(). Examples are given merely to show the use cases.

Parameters:

orig_expr : quantum expression

The original expression which was to be represented

result: Expr

The resulting representation that we wish to integrate over

Examples

>>> from sympy import symbols, DiracDelta
>>> from sympy.physics.quantum.represent import integrate_result
>>> from sympy.physics.quantum.cartesian import XOp, XKet
>>> x_ket = XKet()
>>> X_op = XOp()
>>> x, x_1, x_2 = symbols('x, x_1, x_2')
>>> integrate_result(X_op*x_ket, x*DiracDelta(x-x_1)*DiracDelta(x_1-x_2))
x*DiracDelta(x - x_1)*DiracDelta(x_1 - x_2)
>>> integrate_result(X_op*x_ket, x*DiracDelta(x-x_1)*DiracDelta(x_1-x_2),
...     unities=[1])
x*DiracDelta(x - x_2)

sympy.physics.quantum.represent.get_basis(expr, **options)

Returns a basis state instance corresponding to the basis specified in options=s. If no basis is specified, the function tries to form a default basis state of the given expression.

There are three behaviors:

1. The basis specified in options is already an instance of StateBase. If this is the case, it is simply returned. If the class is specified but not an instance, a default instance is returned.
2. The basis specified is an operator or set of operators. If this is the case, the operator_to_state mapping method is used.
3. No basis is specified. If expr is a state, then a default instance of its class is returned. If expr is an operator, then it is mapped to the corresponding state. If it is neither, then we cannot obtain the basis state.

If the basis cannot be mapped, then it is not changed.

This will be called from within represent, and represent will only pass QExpr’s.

TODO (?): Support for Muls and other types of expressions?

Parameters:

expr : Operator or StateBase

Expression whose basis is sought

Examples

>>> from sympy.physics.quantum.represent import get_basis
>>> from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet
>>> x = XKet()
>>> X = XOp()
>>> get_basis(x)
|x>
>>> get_basis(X)
|x>
>>> get_basis(x, basis=PxOp())
|px>
>>> get_basis(x, basis=PxKet)
|px>

sympy.physics.quantum.represent.enumerate_states(*args, **options)

Returns instances of the given state with dummy indices appended

Operates in two different modes:

1. Two arguments are passed to it. The first is the base state which is to be indexed, and the second argument is a list of indices to append.
2. Three arguments are passed. The first is again the base state to be indexed. The second is the start index for counting. The final argument is the number of kets you wish to receive.

Tries to call state._enumerate_state. If this fails, returns an empty list

Parameters:

args : list

See list of operation modes above for explanation

Examples

>>> from sympy.physics.quantum.cartesian import XBra, XKet
>>> from sympy.physics.quantum.represent import enumerate_states
>>> test = XKet('foo')
>>> enumerate_states(test, 1, 3)
[|foo_1>, |foo_2>, |foo_3>]
>>> test2 = XBra('bar')
>>> enumerate_states(test2, [4, 5, 10])
[<bar_4|, <bar_5|, <bar_10|]