You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
240 lines
7.3 KiB
240 lines
7.3 KiB
"""The commutator: [A,B] = A*B - B*A."""
|
|
|
|
from sympy.core.add import Add
|
|
from sympy.core.expr import Expr
|
|
from sympy.core.mul import Mul
|
|
from sympy.core.power import Pow
|
|
from sympy.core.singleton import S
|
|
from sympy.printing.pretty.stringpict import prettyForm
|
|
|
|
from sympy.physics.quantum.dagger import Dagger
|
|
from sympy.physics.quantum.operator import Operator
|
|
|
|
|
|
__all__ = [
|
|
'Commutator'
|
|
]
|
|
|
|
#-----------------------------------------------------------------------------
|
|
# Commutator
|
|
#-----------------------------------------------------------------------------
|
|
|
|
|
|
class Commutator(Expr):
|
|
"""The standard commutator, in an unevaluated state.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
Evaluating a commutator is defined [1]_ as: ``[A, B] = A*B - B*A``. This
|
|
class returns the commutator in an unevaluated form. To evaluate the
|
|
commutator, use the ``.doit()`` method.
|
|
|
|
Canonical ordering of a commutator is ``[A, B]`` for ``A < B``. The
|
|
arguments of the commutator are put into canonical order using ``__cmp__``.
|
|
If ``B < A``, then ``[B, A]`` is returned as ``-[A, B]``.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
A : Expr
|
|
The first argument of the commutator [A,B].
|
|
B : Expr
|
|
The second argument of the commutator [A,B].
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.physics.quantum import Commutator, Dagger, Operator
|
|
>>> from sympy.abc import x, y
|
|
>>> A = Operator('A')
|
|
>>> B = Operator('B')
|
|
>>> C = Operator('C')
|
|
|
|
Create a commutator and use ``.doit()`` to evaluate it:
|
|
|
|
>>> comm = Commutator(A, B)
|
|
>>> comm
|
|
[A,B]
|
|
>>> comm.doit()
|
|
A*B - B*A
|
|
|
|
The commutator orders it arguments in canonical order:
|
|
|
|
>>> comm = Commutator(B, A); comm
|
|
-[A,B]
|
|
|
|
Commutative constants are factored out:
|
|
|
|
>>> Commutator(3*x*A, x*y*B)
|
|
3*x**2*y*[A,B]
|
|
|
|
Using ``.expand(commutator=True)``, the standard commutator expansion rules
|
|
can be applied:
|
|
|
|
>>> Commutator(A+B, C).expand(commutator=True)
|
|
[A,C] + [B,C]
|
|
>>> Commutator(A, B+C).expand(commutator=True)
|
|
[A,B] + [A,C]
|
|
>>> Commutator(A*B, C).expand(commutator=True)
|
|
[A,C]*B + A*[B,C]
|
|
>>> Commutator(A, B*C).expand(commutator=True)
|
|
[A,B]*C + B*[A,C]
|
|
|
|
Adjoint operations applied to the commutator are properly applied to the
|
|
arguments:
|
|
|
|
>>> Dagger(Commutator(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 S.Zero
|
|
if a.is_commutative or b.is_commutative:
|
|
return S.Zero
|
|
|
|
# [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 in canonical form if A < B.
|
|
if a.compare(b) == 1:
|
|
return S.NegativeOne*cls(b, a)
|
|
|
|
def _expand_pow(self, A, B, sign):
|
|
exp = A.exp
|
|
if not exp.is_integer or not exp.is_constant() or abs(exp) <= 1:
|
|
# nothing to do
|
|
return self
|
|
base = A.base
|
|
if exp.is_negative:
|
|
base = A.base**-1
|
|
exp = -exp
|
|
comm = Commutator(base, B).expand(commutator=True)
|
|
|
|
result = base**(exp - 1) * comm
|
|
for i in range(1, exp):
|
|
result += base**(exp - 1 - i) * comm * base**i
|
|
return sign*result.expand()
|
|
|
|
def _eval_expand_commutator(self, **hints):
|
|
A = self.args[0]
|
|
B = self.args[1]
|
|
|
|
if isinstance(A, Add):
|
|
# [A + B, C] -> [A, C] + [B, C]
|
|
sargs = []
|
|
for term in A.args:
|
|
comm = Commutator(term, B)
|
|
if isinstance(comm, Commutator):
|
|
comm = comm._eval_expand_commutator()
|
|
sargs.append(comm)
|
|
return Add(*sargs)
|
|
elif isinstance(B, Add):
|
|
# [A, B + C] -> [A, B] + [A, C]
|
|
sargs = []
|
|
for term in B.args:
|
|
comm = Commutator(A, term)
|
|
if isinstance(comm, Commutator):
|
|
comm = comm._eval_expand_commutator()
|
|
sargs.append(comm)
|
|
return Add(*sargs)
|
|
elif isinstance(A, Mul):
|
|
# [A*B, C] -> A*[B, C] + [A, C]*B
|
|
a = A.args[0]
|
|
b = Mul(*A.args[1:])
|
|
c = B
|
|
comm1 = Commutator(b, c)
|
|
comm2 = Commutator(a, c)
|
|
if isinstance(comm1, Commutator):
|
|
comm1 = comm1._eval_expand_commutator()
|
|
if isinstance(comm2, Commutator):
|
|
comm2 = comm2._eval_expand_commutator()
|
|
first = Mul(a, comm1)
|
|
second = Mul(comm2, b)
|
|
return Add(first, second)
|
|
elif isinstance(B, Mul):
|
|
# [A, B*C] -> [A, B]*C + B*[A, C]
|
|
a = A
|
|
b = B.args[0]
|
|
c = Mul(*B.args[1:])
|
|
comm1 = Commutator(a, b)
|
|
comm2 = Commutator(a, c)
|
|
if isinstance(comm1, Commutator):
|
|
comm1 = comm1._eval_expand_commutator()
|
|
if isinstance(comm2, Commutator):
|
|
comm2 = comm2._eval_expand_commutator()
|
|
first = Mul(comm1, c)
|
|
second = Mul(b, comm2)
|
|
return Add(first, second)
|
|
elif isinstance(A, Pow):
|
|
# [A**n, C] -> A**(n - 1)*[A, C] + A**(n - 2)*[A, C]*A + ... + [A, C]*A**(n-1)
|
|
return self._expand_pow(A, B, 1)
|
|
elif isinstance(B, Pow):
|
|
# [A, C**n] -> C**(n - 1)*[C, A] + C**(n - 2)*[C, A]*C + ... + [C, A]*C**(n-1)
|
|
return self._expand_pow(B, A, -1)
|
|
|
|
# No changes, so return self
|
|
return self
|
|
|
|
def doit(self, **hints):
|
|
""" Evaluate commutator """
|
|
A = self.args[0]
|
|
B = self.args[1]
|
|
if isinstance(A, Operator) and isinstance(B, Operator):
|
|
try:
|
|
comm = A._eval_commutator(B, **hints)
|
|
except NotImplementedError:
|
|
try:
|
|
comm = -1*B._eval_commutator(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 Commutator(Dagger(self.args[1]), Dagger(self.args[0]))
|
|
|
|
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]" % (
|
|
printer._print(self.args[0]), printer._print(self.args[1]))
|
|
|
|
def _pretty(self, printer, *args):
|
|
pform = printer._print(self.args[0], *args)
|
|
pform = prettyForm(*pform.right(prettyForm(',')))
|
|
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])
|