207 lines
7.0 KiB
Python
207 lines
7.0 KiB
Python
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"""Logic for applying operators to states.
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Todo:
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* Sometimes the final result needs to be expanded, we should do this by hand.
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"""
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from sympy.core.add import Add
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from sympy.core.mul import Mul
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from sympy.core.power import Pow
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from sympy.core.singleton import S
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from sympy.core.sympify import sympify
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from sympy.physics.quantum.anticommutator import AntiCommutator
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from sympy.physics.quantum.commutator import Commutator
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from sympy.physics.quantum.dagger import Dagger
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from sympy.physics.quantum.innerproduct import InnerProduct
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from sympy.physics.quantum.operator import OuterProduct, Operator
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from sympy.physics.quantum.state import State, KetBase, BraBase, Wavefunction
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from sympy.physics.quantum.tensorproduct import TensorProduct
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__all__ = [
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'qapply'
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]
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#-----------------------------------------------------------------------------
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# Main code
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#-----------------------------------------------------------------------------
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def qapply(e, **options):
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"""Apply operators to states in a quantum expression.
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Parameters
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==========
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e : Expr
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The expression containing operators and states. This expression tree
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will be walked to find operators acting on states symbolically.
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options : dict
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A dict of key/value pairs that determine how the operator actions
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are carried out.
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The following options are valid:
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* ``dagger``: try to apply Dagger operators to the left
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(default: False).
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* ``ip_doit``: call ``.doit()`` in inner products when they are
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encountered (default: True).
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Returns
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=======
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e : Expr
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The original expression, but with the operators applied to states.
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Examples
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========
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>>> from sympy.physics.quantum import qapply, Ket, Bra
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>>> b = Bra('b')
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>>> k = Ket('k')
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>>> A = k * b
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>>> A
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|k><b|
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>>> qapply(A * b.dual / (b * b.dual))
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|k>
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>>> qapply(k.dual * A / (k.dual * k), dagger=True)
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<b|
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>>> qapply(k.dual * A / (k.dual * k))
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<k|*|k><b|/<k|k>
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"""
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from sympy.physics.quantum.density import Density
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dagger = options.get('dagger', False)
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if e == 0:
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return S.Zero
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# This may be a bit aggressive but ensures that everything gets expanded
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# to its simplest form before trying to apply operators. This includes
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# things like (A+B+C)*|a> and A*(|a>+|b>) and all Commutators and
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# TensorProducts. The only problem with this is that if we can't apply
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# all the Operators, we have just expanded everything.
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# TODO: don't expand the scalars in front of each Mul.
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e = e.expand(commutator=True, tensorproduct=True)
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# If we just have a raw ket, return it.
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if isinstance(e, KetBase):
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return e
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# We have an Add(a, b, c, ...) and compute
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# Add(qapply(a), qapply(b), ...)
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elif isinstance(e, Add):
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result = 0
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for arg in e.args:
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result += qapply(arg, **options)
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return result.expand()
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# For a Density operator call qapply on its state
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elif isinstance(e, Density):
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new_args = [(qapply(state, **options), prob) for (state,
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prob) in e.args]
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return Density(*new_args)
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# For a raw TensorProduct, call qapply on its args.
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elif isinstance(e, TensorProduct):
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return TensorProduct(*[qapply(t, **options) for t in e.args])
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# For a Pow, call qapply on its base.
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elif isinstance(e, Pow):
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return qapply(e.base, **options)**e.exp
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# We have a Mul where there might be actual operators to apply to kets.
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elif isinstance(e, Mul):
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c_part, nc_part = e.args_cnc()
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c_mul = Mul(*c_part)
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nc_mul = Mul(*nc_part)
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if isinstance(nc_mul, Mul):
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result = c_mul*qapply_Mul(nc_mul, **options)
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else:
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result = c_mul*qapply(nc_mul, **options)
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if result == e and dagger:
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return Dagger(qapply_Mul(Dagger(e), **options))
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else:
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return result
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# In all other cases (State, Operator, Pow, Commutator, InnerProduct,
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# OuterProduct) we won't ever have operators to apply to kets.
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else:
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return e
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def qapply_Mul(e, **options):
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ip_doit = options.get('ip_doit', True)
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args = list(e.args)
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# If we only have 0 or 1 args, we have nothing to do and return.
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if len(args) <= 1 or not isinstance(e, Mul):
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return e
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rhs = args.pop()
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lhs = args.pop()
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# Make sure we have two non-commutative objects before proceeding.
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if (not isinstance(rhs, Wavefunction) and sympify(rhs).is_commutative) or \
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(not isinstance(lhs, Wavefunction) and sympify(lhs).is_commutative):
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return e
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# For a Pow with an integer exponent, apply one of them and reduce the
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# exponent by one.
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if isinstance(lhs, Pow) and lhs.exp.is_Integer:
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args.append(lhs.base**(lhs.exp - 1))
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lhs = lhs.base
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# Pull OuterProduct apart
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if isinstance(lhs, OuterProduct):
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args.append(lhs.ket)
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lhs = lhs.bra
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# Call .doit() on Commutator/AntiCommutator.
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if isinstance(lhs, (Commutator, AntiCommutator)):
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comm = lhs.doit()
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if isinstance(comm, Add):
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return qapply(
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e.func(*(args + [comm.args[0], rhs])) +
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e.func(*(args + [comm.args[1], rhs])),
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**options
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)
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else:
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return qapply(e.func(*args)*comm*rhs, **options)
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# Apply tensor products of operators to states
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if isinstance(lhs, TensorProduct) and all(isinstance(arg, (Operator, State, Mul, Pow)) or arg == 1 for arg in lhs.args) and \
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isinstance(rhs, TensorProduct) and all(isinstance(arg, (Operator, State, Mul, Pow)) or arg == 1 for arg in rhs.args) and \
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len(lhs.args) == len(rhs.args):
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result = TensorProduct(*[qapply(lhs.args[n]*rhs.args[n], **options) for n in range(len(lhs.args))]).expand(tensorproduct=True)
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return qapply_Mul(e.func(*args), **options)*result
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# Now try to actually apply the operator and build an inner product.
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try:
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result = lhs._apply_operator(rhs, **options)
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except (NotImplementedError, AttributeError):
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try:
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result = rhs._apply_from_right_to(lhs, **options)
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except (NotImplementedError, AttributeError):
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if isinstance(lhs, BraBase) and isinstance(rhs, KetBase):
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result = InnerProduct(lhs, rhs)
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if ip_doit:
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result = result.doit()
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else:
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result = None
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# TODO: I may need to expand before returning the final result.
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if result == 0:
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return S.Zero
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elif result is None:
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if len(args) == 0:
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# We had two args to begin with so args=[].
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return e
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else:
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return qapply_Mul(e.func(*(args + [lhs])), **options)*rhs
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elif isinstance(result, InnerProduct):
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return result*qapply_Mul(e.func(*args), **options)
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else: # result is a scalar times a Mul, Add or TensorProduct
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return qapply(e.func(*args)*result, **options)
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