"""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.
"""
from sympy.core.add import Add
from sympy.core.expr import Expr
from sympy.core.function import (Derivative, expand)
from sympy.core.mul import Mul
from sympy.core.numbers import oo
from sympy.core.singleton import S
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.qexpr import QExpr, dispatch_method
from sympy.matrices import eye
__all__ = [
'Operator',
'HermitianOperator',
'UnitaryOperator',
'IdentityOperator',
'OuterProduct',
'DifferentialOperator'
]
#-----------------------------------------------------------------------------
# Operators and outer products
#-----------------------------------------------------------------------------
class Operator(QExpr):
"""Base class for non-commuting quantum operators.
An operator maps between quantum states [1]_. In quantum mechanics,
observables (including, but not limited to, measured physical values) are
represented as Hermitian operators [2]_.
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 do not 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
==========
.. [1] https://en.wikipedia.org/wiki/Operator_%28physics%29
.. [2] https://en.wikipedia.org/wiki/Observable
"""
@classmethod
def default_args(self):
return ("O",)
#-------------------------------------------------------------------------
# Printing
#-------------------------------------------------------------------------
_label_separator = ','
def _print_operator_name(self, printer, *args):
return self.__class__.__name__
_print_operator_name_latex = _print_operator_name
def _print_operator_name_pretty(self, printer, *args):
return prettyForm(self.__class__.__name__)
def _print_contents(self, printer, *args):
if len(self.label) == 1:
return self._print_label(printer, *args)
else:
return '%s(%s)' % (
self._print_operator_name(printer, *args),
self._print_label(printer, *args)
)
def _print_contents_pretty(self, printer, *args):
if len(self.label) == 1:
return self._print_label_pretty(printer, *args)
else:
pform = self._print_operator_name_pretty(printer, *args)
label_pform = self._print_label_pretty(printer, *args)
label_pform = prettyForm(
*label_pform.parens(left='(', right=')')
)
pform = prettyForm(*pform.right(label_pform))
return pform
def _print_contents_latex(self, printer, *args):
if len(self.label) == 1:
return self._print_label_latex(printer, *args)
else:
return r'%s\left(%s\right)' % (
self._print_operator_name_latex(printer, *args),
self._print_label_latex(printer, *args)
)
#-------------------------------------------------------------------------
# _eval_* methods
#-------------------------------------------------------------------------
def _eval_commutator(self, other, **options):
"""Evaluate [self, other] if known, return None if not known."""
return dispatch_method(self, '_eval_commutator', other, **options)
def _eval_anticommutator(self, other, **options):
"""Evaluate [self, other] if known."""
return dispatch_method(self, '_eval_anticommutator', other, **options)
#-------------------------------------------------------------------------
# Operator application
#-------------------------------------------------------------------------
def _apply_operator(self, ket, **options):
return dispatch_method(self, '_apply_operator', ket, **options)
def matrix_element(self, *args):
raise NotImplementedError('matrix_elements is not defined')
def inverse(self):
return self._eval_inverse()
inv = inverse
def _eval_inverse(self):
return self**(-1)
def __mul__(self, other):
if isinstance(other, IdentityOperator):
return self
return Mul(self, other)
class HermitianOperator(Operator):
"""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
"""
is_hermitian = True
def _eval_inverse(self):
if isinstance(self, UnitaryOperator):
return self
else:
return Operator._eval_inverse(self)
def _eval_power(self, exp):
if isinstance(self, UnitaryOperator):
# so all eigenvalues of self are 1 or -1
if exp.is_even:
from sympy.core.singleton import S
return S.One # is identity, see Issue 24153.
elif exp.is_odd:
return self
# No simplification in all other cases
return Operator._eval_power(self, exp)
class UnitaryOperator(Operator):
"""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
"""
def _eval_adjoint(self):
return self._eval_inverse()
class IdentityOperator(Operator):
"""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
"""
@property
def dimension(self):
return self.N
@classmethod
def default_args(self):
return (oo,)
def __init__(self, *args, **hints):
if not len(args) in (0, 1):
raise ValueError('0 or 1 parameters expected, got %s' % args)
self.N = args[0] if (len(args) == 1 and args[0]) else oo
def _eval_commutator(self, other, **hints):
return S.Zero
def _eval_anticommutator(self, other, **hints):
return 2 * other
def _eval_inverse(self):
return self
def _eval_adjoint(self):
return self
def _apply_operator(self, ket, **options):
return ket
def _apply_from_right_to(self, bra, **options):
return bra
def _eval_power(self, exp):
return self
def _print_contents(self, printer, *args):
return 'I'
def _print_contents_pretty(self, printer, *args):
return prettyForm('I')
def _print_contents_latex(self, printer, *args):
return r'{\mathcal{I}}'
def __mul__(self, other):
if isinstance(other, (Operator, Dagger)):
return other
return Mul(self, other)
def _represent_default_basis(self, **options):
if not self.N or self.N == oo:
raise NotImplementedError('Cannot represent infinite dimensional' +
' identity operator as a matrix')
format = options.get('format', 'sympy')
if format != 'sympy':
raise NotImplementedError('Representation in format ' +
'%s not implemented.' % format)
return eye(self.N)
class OuterProduct(Operator):
"""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 [1]_.
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
==========
.. [1] https://en.wikipedia.org/wiki/Outer_product
"""
is_commutative = False
def __new__(cls, *args, **old_assumptions):
from sympy.physics.quantum.state import KetBase, BraBase
if len(args) != 2:
raise ValueError('2 parameters expected, got %d' % len(args))
ket_expr = expand(args[0])
bra_expr = expand(args[1])
if (isinstance(ket_expr, (KetBase, Mul)) and
isinstance(bra_expr, (BraBase, Mul))):
ket_c, kets = ket_expr.args_cnc()
bra_c, bras = bra_expr.args_cnc()
if len(kets) != 1 or not isinstance(kets[0], KetBase):
raise TypeError('KetBase subclass expected'
', got: %r' % Mul(*kets))
if len(bras) != 1 or not isinstance(bras[0], BraBase):
raise TypeError('BraBase subclass expected'
', got: %r' % Mul(*bras))
if not kets[0].dual_class() == bras[0].__class__:
raise TypeError(
'ket and bra are not dual classes: %r, %r' %
(kets[0].__class__, bras[0].__class__)
)
# TODO: make sure the hilbert spaces of the bra and ket are
# compatible
obj = Expr.__new__(cls, *(kets[0], bras[0]), **old_assumptions)
obj.hilbert_space = kets[0].hilbert_space
return Mul(*(ket_c + bra_c)) * obj
op_terms = []
if isinstance(ket_expr, Add) and isinstance(bra_expr, Add):
for ket_term in ket_expr.args:
for bra_term in bra_expr.args:
op_terms.append(OuterProduct(ket_term, bra_term,
**old_assumptions))
elif isinstance(ket_expr, Add):
for ket_term in ket_expr.args:
op_terms.append(OuterProduct(ket_term, bra_expr,
**old_assumptions))
elif isinstance(bra_expr, Add):
for bra_term in bra_expr.args:
op_terms.append(OuterProduct(ket_expr, bra_term,
**old_assumptions))
else:
raise TypeError(
'Expected ket and bra expression, got: %r, %r' %
(ket_expr, bra_expr)
)
return Add(*op_terms)
@property
def ket(self):
"""Return the ket on the left side of the outer product."""
return self.args[0]
@property
def bra(self):
"""Return the bra on the right side of the outer product."""
return self.args[1]
def _eval_adjoint(self):
return OuterProduct(Dagger(self.bra), Dagger(self.ket))
def _sympystr(self, printer, *args):
return printer._print(self.ket) + printer._print(self.bra)
def _sympyrepr(self, printer, *args):
return '%s(%s,%s)' % (self.__class__.__name__,
printer._print(self.ket, *args), printer._print(self.bra, *args))
def _pretty(self, printer, *args):
pform = self.ket._pretty(printer, *args)
return prettyForm(*pform.right(self.bra._pretty(printer, *args)))
def _latex(self, printer, *args):
k = printer._print(self.ket, *args)
b = printer._print(self.bra, *args)
return k + b
def _represent(self, **options):
k = self.ket._represent(**options)
b = self.bra._represent(**options)
return k*b
def _eval_trace(self, **kwargs):
# TODO if operands are tensorproducts this may be will be handled
# differently.
return self.ket._eval_trace(self.bra, **kwargs)
class DifferentialOperator(Operator):
"""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
def variables(self):
"""
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)
"""
return self.args[-1].args
@property
def function(self):
"""
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)
"""
return self.args[-1]
@property
def expr(self):
"""
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)
"""
return self.args[0]
@property
def free_symbols(self):
"""
Return the free symbols of the expression.
"""
return self.expr.free_symbols
def _apply_operator_Wavefunction(self, func, **options):
from sympy.physics.quantum.state import Wavefunction
var = self.variables
wf_vars = func.args[1:]
f = self.function
new_expr = self.expr.subs(f, func(*var))
new_expr = new_expr.doit()
return Wavefunction(new_expr, *wf_vars)
def _eval_derivative(self, symbol):
new_expr = Derivative(self.expr, symbol)
return DifferentialOperator(new_expr, self.args[-1])
#-------------------------------------------------------------------------
# Printing
#-------------------------------------------------------------------------
def _print(self, printer, *args):
return '%s(%s)' % (
self._print_operator_name(printer, *args),
self._print_label(printer, *args)
)
def _print_pretty(self, printer, *args):
pform = self._print_operator_name_pretty(printer, *args)
label_pform = self._print_label_pretty(printer, *args)
label_pform = prettyForm(
*label_pform.parens(left='(', right=')')
)
pform = prettyForm(*pform.right(label_pform))
return pform