"""The anti-commutator: ``{A,B} = A*B + B*A``."""
from __future__ import print_function, division
from sympy import S, Expr, Mul, Integer
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.operator import Operator
from sympy.physics.quantum.dagger import Dagger
__all__ = [
'AntiCommutator'
]
#-----------------------------------------------------------------------------
# Anti-commutator
#-----------------------------------------------------------------------------
[docs]class AntiCommutator(Expr):
"""The standard anticommutator, in an unevaluated state.
Evaluating an anticommutator is defined [1]_ as: ``{A, B} = A*B + B*A``.
This class returns the anticommutator in an unevaluated form. To evaluate
the anticommutator, use the ``.doit()`` method.
Canonical ordering of an anticommutator is ``{A, B}`` for ``A < B``. The
arguments of the anticommutator are put into canonical order using
``__cmp__``. If ``B < A``, then ``{A, B}`` is returned as ``{B, A}``.
Parameters
==========
A : Expr
The first argument of the anticommutator {A,B}.
B : Expr
The second argument of the anticommutator {A,B}.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.quantum import AntiCommutator
>>> from sympy.physics.quantum import Operator, Dagger
>>> x, y = symbols('x,y')
>>> A = Operator('A')
>>> B = Operator('B')
Create an anticommutator and use ``doit()`` to multiply them out.
>>> ac = AntiCommutator(A,B); ac
{A,B}
>>> ac.doit()
A*B + B*A
The commutator orders it arguments in canonical order:
>>> ac = AntiCommutator(B,A); ac
{A,B}
Commutative constants are factored out:
>>> AntiCommutator(3*x*A,x*y*B)
3*x**2*y*{A,B}
Adjoint operations applied to the anticommutator are properly applied to
the arguments:
>>> Dagger(AntiCommutator(A,B))
{Dagger(A),Dagger(B)}
References
==========
.. [1] https://en.wikipedia.org/wiki/Commutator
"""
is_commutative = False
def __new__(cls, A, B):
r = cls.eval(A, B)
if r is not None:
return r
obj = Expr.__new__(cls, A, B)
return obj
@classmethod
def eval(cls, a, b):
if not (a and b):
return S.Zero
if a == b:
return Integer(2)*a**2
if a.is_commutative or b.is_commutative:
return Integer(2)*a*b
# [xA,yB] -> xy*[A,B]
ca, nca = a.args_cnc()
cb, ncb = b.args_cnc()
c_part = ca + cb
if c_part:
return Mul(Mul(*c_part), cls(Mul._from_args(nca), Mul._from_args(ncb)))
# Canonical ordering of arguments
#The Commutator [A,B] is on canonical form if A < B.
if a.compare(b) == 1:
return cls(b, a)
[docs] def doit(self, **hints):
""" Evaluate anticommutator """
A = self.args[0]
B = self.args[1]
if isinstance(A, Operator) and isinstance(B, Operator):
try:
comm = A._eval_anticommutator(B, **hints)
except NotImplementedError:
try:
comm = B._eval_anticommutator(A, **hints)
except NotImplementedError:
comm = None
if comm is not None:
return comm.doit(**hints)
return (A*B + B*A).doit(**hints)
def _eval_adjoint(self):
return AntiCommutator(Dagger(self.args[0]), Dagger(self.args[1]))
def _sympyrepr(self, printer, *args):
return "%s(%s,%s)" % (
self.__class__.__name__, printer._print(
self.args[0]), printer._print(self.args[1])
)
def _sympystr(self, printer, *args):
return "{%s,%s}" % (self.args[0], self.args[1])
def _pretty(self, printer, *args):
pform = printer._print(self.args[0], *args)
pform = prettyForm(*pform.right((prettyForm(u','))))
pform = prettyForm(*pform.right((printer._print(self.args[1], *args))))
pform = prettyForm(*pform.parens(left='{', right='}'))
return pform
def _latex(self, printer, *args):
return "\\left\\{%s,%s\\right\\}" % tuple([
printer._print(arg, *args) for arg in self.args])
