"""Logic for representing operators in state in various bases.
TODO:
* Get represent working with continuous hilbert spaces.
* Document default basis functionality.
"""
from sympy.core.add import Add
from sympy.core.expr import Expr
from sympy.core.mul import Mul
from sympy.core.numbers import I
from sympy.core.power import Pow
from sympy.integrals.integrals import integrate
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.commutator import Commutator
from sympy.physics.quantum.anticommutator import AntiCommutator
from sympy.physics.quantum.innerproduct import InnerProduct
from sympy.physics.quantum.qexpr import QExpr
from sympy.physics.quantum.tensorproduct import TensorProduct
from sympy.physics.quantum.matrixutils import flatten_scalar
from sympy.physics.quantum.state import KetBase, BraBase, StateBase
from sympy.physics.quantum.operator import Operator, OuterProduct
from sympy.physics.quantum.qapply import qapply
from sympy.physics.quantum.operatorset import operators_to_state, state_to_operators
__all__ = [
'represent',
'rep_innerproduct',
'rep_expectation',
'integrate_result',
'get_basis',
'enumerate_states'
]
#-----------------------------------------------------------------------------
# Represent
#-----------------------------------------------------------------------------
def _sympy_to_scalar(e):
"""Convert from a SymPy scalar to a Python scalar."""
if isinstance(e, Expr):
if e.is_Integer:
return int(e)
elif e.is_Float:
return float(e)
elif e.is_Rational:
return float(e)
elif e.is_Number or e.is_NumberSymbol or e == I:
return complex(e)
raise TypeError('Expected number, got: %r' % e)
def 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 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)
"""
format = options.get('format', 'sympy')
if format == 'numpy':
import numpy as np
if isinstance(expr, QExpr) and not isinstance(expr, OuterProduct):
options['replace_none'] = False
temp_basis = get_basis(expr, **options)
if temp_basis is not None:
options['basis'] = temp_basis
try:
return expr._represent(**options)
except NotImplementedError as strerr:
#If no _represent_FOO method exists, map to the
#appropriate basis state and try
#the other methods of representation
options['replace_none'] = True
if isinstance(expr, (KetBase, BraBase)):
try:
return rep_innerproduct(expr, **options)
except NotImplementedError:
raise NotImplementedError(strerr)
elif isinstance(expr, Operator):
try:
return rep_expectation(expr, **options)
except NotImplementedError:
raise NotImplementedError(strerr)
else:
raise NotImplementedError(strerr)
elif isinstance(expr, Add):
result = represent(expr.args[0], **options)
for args in expr.args[1:]:
# scipy.sparse doesn't support += so we use plain = here.
result = result + represent(args, **options)
return result
elif isinstance(expr, Pow):
base, exp = expr.as_base_exp()
if format in ('numpy', 'scipy.sparse'):
exp = _sympy_to_scalar(exp)
base = represent(base, **options)
# scipy.sparse doesn't support negative exponents
# and warns when inverting a matrix in csr format.
if format == 'scipy.sparse' and exp < 0:
from scipy.sparse.linalg import inv
exp = - exp
base = inv(base.tocsc()).tocsr()
if format == 'numpy':
return np.linalg.matrix_power(base, exp)
return base ** exp
elif isinstance(expr, TensorProduct):
new_args = [represent(arg, **options) for arg in expr.args]
return TensorProduct(*new_args)
elif isinstance(expr, Dagger):
return Dagger(represent(expr.args[0], **options))
elif isinstance(expr, Commutator):
A = expr.args[0]
B = expr.args[1]
return represent(Mul(A, B) - Mul(B, A), **options)
elif isinstance(expr, AntiCommutator):
A = expr.args[0]
B = expr.args[1]
return represent(Mul(A, B) + Mul(B, A), **options)
elif isinstance(expr, InnerProduct):
return represent(Mul(expr.bra, expr.ket), **options)
elif not isinstance(expr, (Mul, OuterProduct)):
# For numpy and scipy.sparse, we can only handle numerical prefactors.
if format in ('numpy', 'scipy.sparse'):
return _sympy_to_scalar(expr)
return expr
if not isinstance(expr, (Mul, OuterProduct)):
raise TypeError('Mul expected, got: %r' % expr)
if "index" in options:
options["index"] += 1
else:
options["index"] = 1
if "unities" not in options:
options["unities"] = []
result = represent(expr.args[-1], **options)
last_arg = expr.args[-1]
for arg in reversed(expr.args[:-1]):
if isinstance(last_arg, Operator):
options["index"] += 1
options["unities"].append(options["index"])
elif isinstance(last_arg, BraBase) and isinstance(arg, KetBase):
options["index"] += 1
elif isinstance(last_arg, KetBase) and isinstance(arg, Operator):
options["unities"].append(options["index"])
elif isinstance(last_arg, KetBase) and isinstance(arg, BraBase):
options["unities"].append(options["index"])
next_arg = represent(arg, **options)
if format == 'numpy' and isinstance(next_arg, np.ndarray):
# Must use np.matmult to "matrix multiply" two np.ndarray
result = np.matmul(next_arg, result)
else:
result = next_arg*result
last_arg = arg
# All three matrix formats create 1 by 1 matrices when inner products of
# vectors are taken. In these cases, we simply return a scalar.
result = flatten_scalar(result)
result = integrate_result(expr, result, **options)
return result
def 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))
"""
if not isinstance(expr, (KetBase, BraBase)):
raise TypeError("expr passed is not a Bra or Ket")
basis = get_basis(expr, **options)
if not isinstance(basis, StateBase):
raise NotImplementedError("Can't form this representation!")
if "index" not in options:
options["index"] = 1
basis_kets = enumerate_states(basis, options["index"], 2)
if isinstance(expr, BraBase):
bra = expr
ket = (basis_kets[1] if basis_kets[0].dual == expr else basis_kets[0])
else:
bra = (basis_kets[1].dual if basis_kets[0]
== expr else basis_kets[0].dual)
ket = expr
prod = InnerProduct(bra, ket)
result = prod.doit()
format = options.get('format', 'sympy')
return expr._format_represent(result, format)
def 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, 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>
"""
if "index" not in options:
options["index"] = 1
if not isinstance(expr, Operator):
raise TypeError("The passed expression is not an operator")
basis_state = get_basis(expr, **options)
if basis_state is None or not isinstance(basis_state, StateBase):
raise NotImplementedError("Could not get basis kets for this operator")
basis_kets = enumerate_states(basis_state, options["index"], 2)
bra = basis_kets[1].dual
ket = basis_kets[0]
return qapply(bra*expr*ket)
def 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)
"""
if not isinstance(result, Expr):
return result
options['replace_none'] = True
if "basis" not in options:
arg = orig_expr.args[-1]
options["basis"] = get_basis(arg, **options)
elif not isinstance(options["basis"], StateBase):
options["basis"] = get_basis(orig_expr, **options)
basis = options.pop("basis", None)
if basis is None:
return result
unities = options.pop("unities", [])
if len(unities) == 0:
return result
kets = enumerate_states(basis, unities)
coords = [k.label[0] for k in kets]
for coord in coords:
if coord in result.free_symbols:
#TODO: Add support for sets of operators
basis_op = state_to_operators(basis)
start = basis_op.hilbert_space.interval.start
end = basis_op.hilbert_space.interval.end
result = integrate(result, (coord, start, end))
return result
def get_basis(expr, *, basis=None, replace_none=True, **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>
"""
if basis is None and not replace_none:
return None
if basis is None:
if isinstance(expr, KetBase):
return _make_default(expr.__class__)
elif isinstance(expr, BraBase):
return _make_default(expr.dual_class())
elif isinstance(expr, Operator):
state_inst = operators_to_state(expr)
return (state_inst if state_inst is not None else None)
else:
return None
elif (isinstance(basis, Operator) or
(not isinstance(basis, StateBase) and issubclass(basis, Operator))):
state = operators_to_state(basis)
if state is None:
return None
elif isinstance(state, StateBase):
return state
else:
return _make_default(state)
elif isinstance(basis, StateBase):
return basis
elif issubclass(basis, StateBase):
return _make_default(basis)
else:
return None
def _make_default(expr):
# XXX: Catching TypeError like this is a bad way of distinguishing
# instances from classes. The logic using this function should be
# rewritten somehow.
try:
expr = expr()
except TypeError:
return expr
return expr
def 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|]
"""
state = args[0]
if len(args) not in (2, 3):
raise NotImplementedError("Wrong number of arguments!")
if not isinstance(state, StateBase):
raise TypeError("First argument is not a state!")
if len(args) == 3:
num_states = args[2]
options['start_index'] = args[1]
else:
num_states = len(args[1])
options['index_list'] = args[1]
try:
ret = state._enumerate_state(num_states, **options)
except NotImplementedError:
ret = []
return ret