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"])
result = represent(arg, **options)*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
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"])
result = represent(arg, **options) * 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
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 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 == 'numpy' or format == 'scipy.sparse':
exp = _sympy_to_scalar(exp)
return represent(base, **options)**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 = represent(expr.args[0], **options)
B = represent(expr.args[1], **options)
return A * B - B * A
elif isinstance(expr, AntiCommutator):
A = represent(expr.args[0], **options)
B = represent(expr.args[1], **options)
return A * B + B * A
elif isinstance(expr, InnerProduct):
return represent(Mul(expr.bra, expr.ket), **options)
elif not (isinstance(expr, Mul) or isinstance(expr, OuterProduct)):
# For numpy and scipy.sparse, we can only handle numerical prefactors.
if format == 'numpy' or format == 'scipy.sparse':
return _sympy_to_scalar(expr)
return expr
if not (isinstance(expr, Mul) or isinstance(expr, OuterProduct)):
raise TypeError('Mul expected, got: %r' % expr)
if "index" in options:
options["index"] += 1
else:
options["index"] = 1
if not "unities" 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"])
result = represent(arg, **options) * 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 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 does 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)
[1]
[0]
"""
format = options.get("format", "sympy")
if isinstance(expr, QExpr):
return expr._represent(**options)
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):
exp = expr.exp
if format == "numpy" or format == "scipy.sparse":
exp = _sympy_to_scalar(exp)
return represent(expr.base, **options) ** 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 = represent(expr.args[0], **options)
B = represent(expr.args[1], **options)
return A * B - B * A
elif isinstance(expr, AntiCommutator):
A = represent(expr.args[0], **options)
B = represent(expr.args[1], **options)
return A * B + B * A
elif isinstance(expr, InnerProduct):
return represent(Mul(expr.bra, expr.ket), **options)
elif not isinstance(expr, Mul):
# For numpy and scipy.sparse, we can only handle numerical prefactors.
if format == "numpy" or format == "scipy.sparse":
return _sympy_to_scalar(expr)
return expr
if not isinstance(expr, Mul):
raise TypeError("Mul expected, got: %r" % expr)
result = represent(expr.args[-1], **options)
for arg in reversed(expr.args[:-1]):
result = represent(arg, **options) * result
# 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)
return result
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 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 == 'numpy' or format == 'scipy.sparse':
exp = _sympy_to_scalar(exp)
return represent(base, **options)**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 = represent(expr.args[0], **options)
B = represent(expr.args[1], **options)
return A*B - B*A
elif isinstance(expr, AntiCommutator):
A = represent(expr.args[0], **options)
B = represent(expr.args[1], **options)
return A*B + B*A
elif isinstance(expr, InnerProduct):
return represent(Mul(expr.bra, expr.ket), **options)
elif not (isinstance(expr, Mul) or isinstance(expr, OuterProduct)):
# For numpy and scipy.sparse, we can only handle numerical prefactors.
if format == 'numpy' or format == 'scipy.sparse':
return _sympy_to_scalar(expr)
return expr
if not (isinstance(expr, Mul) or isinstance(expr, OuterProduct)):
raise TypeError('Mul expected, got: %r' % expr)
if "index" in options:
options["index"] += 1
else:
options["index"] = 1
if not "unities" 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"])
result = represent(arg, **options)*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 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 does 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)
[1]
[0]
"""
format = options.get('format', 'sympy')
if isinstance(expr, QExpr):
return expr._represent(**options)
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):
exp = expr.exp
if format == 'numpy' or format == 'scipy.sparse':
exp = _sympy_to_scalar(exp)
return represent(expr.base, **options)**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 = represent(expr.args[0], **options)
B = represent(expr.args[1], **options)
return A * B - B * A
elif isinstance(expr, AntiCommutator):
A = represent(expr.args[0], **options)
B = represent(expr.args[1], **options)
return A * B + B * A
elif isinstance(expr, InnerProduct):
return represent(Mul(expr.bra, expr.ket), **options)
elif not isinstance(expr, Mul):
# For numpy and scipy.sparse, we can only handle numerical prefactors.
if format == 'numpy' or format == 'scipy.sparse':
return _sympy_to_scalar(expr)
return expr
if not isinstance(expr, Mul):
raise TypeError('Mul expected, got: %r' % expr)
result = represent(expr.args[-1], **options)
for arg in reversed(expr.args[:-1]):
result = represent(arg, **options) * result
# 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)
return result
