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.enumerate_states(*args, **options)[source]¶
Returns instances of the given state with dummy indices appended
Operates in two different modes:
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.
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|]
- sympy.physics.quantum.represent.get_basis(expr, *, basis=None, replace_none=True, **options)[source]¶
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:
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.
The basis specified is an operator or set of operators. If this is the case, the operator_to_state mapping method is used.
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.integrate_result(orig_expr, result, **options)[source]¶
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.rep_expectation(expr, **options)[source]¶
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, 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.rep_innerproduct(expr, **options)[source]¶
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.represent(expr, **options)[source]¶
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
andKet
to create the z-spin operator and its spin 1/2 up eigenstate. By defining 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)