575 lines
18 KiB
Python
575 lines
18 KiB
Python
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"""Logic for representing operators in state in various bases.
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TODO:
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* Get represent working with continuous hilbert spaces.
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* Document default basis functionality.
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"""
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from sympy.core.add import Add
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from sympy.core.expr import Expr
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from sympy.core.mul import Mul
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from sympy.core.numbers import I
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from sympy.core.power import Pow
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from sympy.integrals.integrals import integrate
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from sympy.physics.quantum.dagger import Dagger
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from sympy.physics.quantum.commutator import Commutator
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from sympy.physics.quantum.anticommutator import AntiCommutator
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from sympy.physics.quantum.innerproduct import InnerProduct
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from sympy.physics.quantum.qexpr import QExpr
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from sympy.physics.quantum.tensorproduct import TensorProduct
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from sympy.physics.quantum.matrixutils import flatten_scalar
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from sympy.physics.quantum.state import KetBase, BraBase, StateBase
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from sympy.physics.quantum.operator import Operator, OuterProduct
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from sympy.physics.quantum.qapply import qapply
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from sympy.physics.quantum.operatorset import operators_to_state, state_to_operators
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__all__ = [
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'represent',
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'rep_innerproduct',
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'rep_expectation',
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'integrate_result',
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'get_basis',
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'enumerate_states'
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]
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#-----------------------------------------------------------------------------
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# Represent
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#-----------------------------------------------------------------------------
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def _sympy_to_scalar(e):
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"""Convert from a SymPy scalar to a Python scalar."""
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if isinstance(e, Expr):
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if e.is_Integer:
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return int(e)
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elif e.is_Float:
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return float(e)
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elif e.is_Rational:
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return float(e)
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elif e.is_Number or e.is_NumberSymbol or e == I:
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return complex(e)
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raise TypeError('Expected number, got: %r' % e)
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def represent(expr, **options):
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"""Represent the quantum expression in the given basis.
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In quantum mechanics abstract states and operators can be represented in
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various basis sets. Under this operation the follow transforms happen:
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* Ket -> column vector or function
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* Bra -> row vector of function
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* Operator -> matrix or differential operator
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This function is the top-level interface for this action.
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This function walks the SymPy expression tree looking for ``QExpr``
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instances that have a ``_represent`` method. This method is then called
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and the object is replaced by the representation returned by this method.
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By default, the ``_represent`` method will dispatch to other methods
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that handle the representation logic for a particular basis set. The
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naming convention for these methods is the following::
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def _represent_FooBasis(self, e, basis, **options)
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This function will have the logic for representing instances of its class
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in the basis set having a class named ``FooBasis``.
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Parameters
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==========
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expr : Expr
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The expression to represent.
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basis : Operator, basis set
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An object that contains the information about the basis set. If an
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operator is used, the basis is assumed to be the orthonormal
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eigenvectors of that operator. In general though, the basis argument
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can be any object that contains the basis set information.
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options : dict
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Key/value pairs of options that are passed to the underlying method
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that finds the representation. These options can be used to
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control how the representation is done. For example, this is where
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the size of the basis set would be set.
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Returns
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=======
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e : Expr
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The SymPy expression of the represented quantum expression.
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Examples
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========
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Here we subclass ``Operator`` and ``Ket`` to create the z-spin operator
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and its spin 1/2 up eigenstate. By defining the ``_represent_SzOp``
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method, the ket can be represented in the z-spin basis.
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>>> from sympy.physics.quantum import Operator, represent, Ket
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>>> from sympy import Matrix
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>>> class SzUpKet(Ket):
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... def _represent_SzOp(self, basis, **options):
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... return Matrix([1,0])
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...
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>>> class SzOp(Operator):
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... pass
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...
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>>> sz = SzOp('Sz')
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>>> up = SzUpKet('up')
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>>> represent(up, basis=sz)
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Matrix([
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[1],
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[0]])
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Here we see an example of representations in a continuous
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basis. We see that the result of representing various combinations
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of cartesian position operators and kets give us continuous
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expressions involving DiracDelta functions.
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>>> from sympy.physics.quantum.cartesian import XOp, XKet, XBra
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>>> X = XOp()
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>>> x = XKet()
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>>> y = XBra('y')
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>>> represent(X*x)
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x*DiracDelta(x - x_2)
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>>> represent(X*x*y)
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x*DiracDelta(x - x_3)*DiracDelta(x_1 - y)
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"""
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format = options.get('format', 'sympy')
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if format == 'numpy':
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import numpy as np
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if isinstance(expr, QExpr) and not isinstance(expr, OuterProduct):
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options['replace_none'] = False
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temp_basis = get_basis(expr, **options)
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if temp_basis is not None:
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options['basis'] = temp_basis
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try:
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return expr._represent(**options)
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except NotImplementedError as strerr:
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#If no _represent_FOO method exists, map to the
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#appropriate basis state and try
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#the other methods of representation
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options['replace_none'] = True
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if isinstance(expr, (KetBase, BraBase)):
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try:
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return rep_innerproduct(expr, **options)
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except NotImplementedError:
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raise NotImplementedError(strerr)
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elif isinstance(expr, Operator):
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try:
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return rep_expectation(expr, **options)
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except NotImplementedError:
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raise NotImplementedError(strerr)
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else:
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raise NotImplementedError(strerr)
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elif isinstance(expr, Add):
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result = represent(expr.args[0], **options)
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for args in expr.args[1:]:
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# scipy.sparse doesn't support += so we use plain = here.
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result = result + represent(args, **options)
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return result
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elif isinstance(expr, Pow):
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base, exp = expr.as_base_exp()
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if format in ('numpy', 'scipy.sparse'):
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exp = _sympy_to_scalar(exp)
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base = represent(base, **options)
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# scipy.sparse doesn't support negative exponents
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# and warns when inverting a matrix in csr format.
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if format == 'scipy.sparse' and exp < 0:
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from scipy.sparse.linalg import inv
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exp = - exp
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base = inv(base.tocsc()).tocsr()
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if format == 'numpy':
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return np.linalg.matrix_power(base, exp)
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return base ** exp
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elif isinstance(expr, TensorProduct):
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new_args = [represent(arg, **options) for arg in expr.args]
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return TensorProduct(*new_args)
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elif isinstance(expr, Dagger):
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return Dagger(represent(expr.args[0], **options))
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elif isinstance(expr, Commutator):
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A = expr.args[0]
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B = expr.args[1]
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return represent(Mul(A, B) - Mul(B, A), **options)
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elif isinstance(expr, AntiCommutator):
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A = expr.args[0]
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B = expr.args[1]
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return represent(Mul(A, B) + Mul(B, A), **options)
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elif isinstance(expr, InnerProduct):
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return represent(Mul(expr.bra, expr.ket), **options)
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elif not isinstance(expr, (Mul, OuterProduct)):
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# For numpy and scipy.sparse, we can only handle numerical prefactors.
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if format in ('numpy', 'scipy.sparse'):
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return _sympy_to_scalar(expr)
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return expr
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if not isinstance(expr, (Mul, OuterProduct)):
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raise TypeError('Mul expected, got: %r' % expr)
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if "index" in options:
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options["index"] += 1
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else:
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options["index"] = 1
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if "unities" not in options:
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options["unities"] = []
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result = represent(expr.args[-1], **options)
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last_arg = expr.args[-1]
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for arg in reversed(expr.args[:-1]):
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if isinstance(last_arg, Operator):
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options["index"] += 1
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options["unities"].append(options["index"])
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elif isinstance(last_arg, BraBase) and isinstance(arg, KetBase):
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options["index"] += 1
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elif isinstance(last_arg, KetBase) and isinstance(arg, Operator):
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options["unities"].append(options["index"])
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elif isinstance(last_arg, KetBase) and isinstance(arg, BraBase):
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options["unities"].append(options["index"])
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next_arg = represent(arg, **options)
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if format == 'numpy' and isinstance(next_arg, np.ndarray):
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# Must use np.matmult to "matrix multiply" two np.ndarray
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result = np.matmul(next_arg, result)
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else:
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result = next_arg*result
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last_arg = arg
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# All three matrix formats create 1 by 1 matrices when inner products of
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# vectors are taken. In these cases, we simply return a scalar.
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result = flatten_scalar(result)
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result = integrate_result(expr, result, **options)
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return result
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def rep_innerproduct(expr, **options):
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"""
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Returns an innerproduct like representation (e.g. ``<x'|x>``) for the
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given state.
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Attempts to calculate inner product with a bra from the specified
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basis. Should only be passed an instance of KetBase or BraBase
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Parameters
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==========
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expr : KetBase or BraBase
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The expression to be represented
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Examples
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========
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>>> from sympy.physics.quantum.represent import rep_innerproduct
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>>> from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet
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>>> rep_innerproduct(XKet())
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DiracDelta(x - x_1)
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>>> rep_innerproduct(XKet(), basis=PxOp())
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sqrt(2)*exp(-I*px_1*x/hbar)/(2*sqrt(hbar)*sqrt(pi))
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>>> rep_innerproduct(PxKet(), basis=XOp())
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sqrt(2)*exp(I*px*x_1/hbar)/(2*sqrt(hbar)*sqrt(pi))
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"""
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if not isinstance(expr, (KetBase, BraBase)):
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raise TypeError("expr passed is not a Bra or Ket")
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basis = get_basis(expr, **options)
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if not isinstance(basis, StateBase):
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raise NotImplementedError("Can't form this representation!")
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if "index" not in options:
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options["index"] = 1
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basis_kets = enumerate_states(basis, options["index"], 2)
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if isinstance(expr, BraBase):
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bra = expr
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ket = (basis_kets[1] if basis_kets[0].dual == expr else basis_kets[0])
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else:
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bra = (basis_kets[1].dual if basis_kets[0]
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== expr else basis_kets[0].dual)
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ket = expr
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prod = InnerProduct(bra, ket)
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result = prod.doit()
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format = options.get('format', 'sympy')
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return expr._format_represent(result, format)
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def rep_expectation(expr, **options):
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"""
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Returns an ``<x'|A|x>`` type representation for the given operator.
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Parameters
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==========
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expr : Operator
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Operator to be represented in the specified basis
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Examples
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========
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>>> from sympy.physics.quantum.cartesian import XOp, PxOp, PxKet
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>>> from sympy.physics.quantum.represent import rep_expectation
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>>> rep_expectation(XOp())
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x_1*DiracDelta(x_1 - x_2)
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>>> rep_expectation(XOp(), basis=PxOp())
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<px_2|*X*|px_1>
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>>> rep_expectation(XOp(), basis=PxKet())
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<px_2|*X*|px_1>
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"""
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if "index" not in options:
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options["index"] = 1
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if not isinstance(expr, Operator):
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raise TypeError("The passed expression is not an operator")
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basis_state = get_basis(expr, **options)
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if basis_state is None or not isinstance(basis_state, StateBase):
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raise NotImplementedError("Could not get basis kets for this operator")
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basis_kets = enumerate_states(basis_state, options["index"], 2)
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bra = basis_kets[1].dual
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ket = basis_kets[0]
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return qapply(bra*expr*ket)
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def integrate_result(orig_expr, result, **options):
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"""
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Returns the result of integrating over any unities ``(|x><x|)`` in
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the given expression. Intended for integrating over the result of
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representations in continuous bases.
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This function integrates over any unities that may have been
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inserted into the quantum expression and returns the result.
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It uses the interval of the Hilbert space of the basis state
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passed to it in order to figure out the limits of integration.
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The unities option must be
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specified for this to work.
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Note: This is mostly used internally by represent(). Examples are
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given merely to show the use cases.
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Parameters
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==========
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orig_expr : quantum expression
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The original expression which was to be represented
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result: Expr
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The resulting representation that we wish to integrate over
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Examples
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========
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>>> from sympy import symbols, DiracDelta
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>>> from sympy.physics.quantum.represent import integrate_result
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>>> from sympy.physics.quantum.cartesian import XOp, XKet
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>>> x_ket = XKet()
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||
|
|
>>> X_op = XOp()
|
||
|
|
>>> x, x_1, x_2 = symbols('x, x_1, x_2')
|
||
|
|
>>> integrate_result(X_op*x_ket, x*DiracDelta(x-x_1)*DiracDelta(x_1-x_2))
|
||
|
|
x*DiracDelta(x - x_1)*DiracDelta(x_1 - x_2)
|
||
|
|
>>> integrate_result(X_op*x_ket, x*DiracDelta(x-x_1)*DiracDelta(x_1-x_2),
|
||
|
|
... unities=[1])
|
||
|
|
x*DiracDelta(x - x_2)
|
||
|
|
|
||
|
|
"""
|
||
|
|
if not isinstance(result, Expr):
|
||
|
|
return result
|
||
|
|
|
||
|
|
options['replace_none'] = True
|
||
|
|
if "basis" not in options:
|
||
|
|
arg = orig_expr.args[-1]
|
||
|
|
options["basis"] = get_basis(arg, **options)
|
||
|
|
elif not isinstance(options["basis"], StateBase):
|
||
|
|
options["basis"] = get_basis(orig_expr, **options)
|
||
|
|
|
||
|
|
basis = options.pop("basis", None)
|
||
|
|
|
||
|
|
if basis is None:
|
||
|
|
return result
|
||
|
|
|
||
|
|
unities = options.pop("unities", [])
|
||
|
|
|
||
|
|
if len(unities) == 0:
|
||
|
|
return result
|
||
|
|
|
||
|
|
kets = enumerate_states(basis, unities)
|
||
|
|
coords = [k.label[0] for k in kets]
|
||
|
|
|
||
|
|
for coord in coords:
|
||
|
|
if coord in result.free_symbols:
|
||
|
|
#TODO: Add support for sets of operators
|
||
|
|
basis_op = state_to_operators(basis)
|
||
|
|
start = basis_op.hilbert_space.interval.start
|
||
|
|
end = basis_op.hilbert_space.interval.end
|
||
|
|
result = integrate(result, (coord, start, end))
|
||
|
|
|
||
|
|
return result
|
||
|
|
|
||
|
|
|
||
|
|
def get_basis(expr, *, basis=None, replace_none=True, **options):
|
||
|
|
"""
|
||
|
|
Returns a basis state instance corresponding to the basis specified in
|
||
|
|
options=s. If no basis is specified, the function tries to form a default
|
||
|
|
basis state of the given expression.
|
||
|
|
|
||
|
|
There are three behaviors:
|
||
|
|
|
||
|
|
1. The basis specified in options is already an instance of StateBase. If
|
||
|
|
this is the case, it is simply returned. If the class is specified but
|
||
|
|
not an instance, a default instance is returned.
|
||
|
|
|
||
|
|
2. The basis specified is an operator or set of operators. If this
|
||
|
|
is the case, the operator_to_state mapping method is used.
|
||
|
|
|
||
|
|
3. No basis is specified. If expr is a state, then a default instance of
|
||
|
|
its class is returned. If expr is an operator, then it is mapped to the
|
||
|
|
corresponding state. If it is neither, then we cannot obtain the basis
|
||
|
|
state.
|
||
|
|
|
||
|
|
If the basis cannot be mapped, then it is not changed.
|
||
|
|
|
||
|
|
This will be called from within represent, and represent will
|
||
|
|
only pass QExpr's.
|
||
|
|
|
||
|
|
TODO (?): Support for Muls and other types of expressions?
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
expr : Operator or StateBase
|
||
|
|
Expression whose basis is sought
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.represent import get_basis
|
||
|
|
>>> from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet
|
||
|
|
>>> x = XKet()
|
||
|
|
>>> X = XOp()
|
||
|
|
>>> get_basis(x)
|
||
|
|
|x>
|
||
|
|
>>> get_basis(X)
|
||
|
|
|x>
|
||
|
|
>>> get_basis(x, basis=PxOp())
|
||
|
|
|px>
|
||
|
|
>>> get_basis(x, basis=PxKet)
|
||
|
|
|px>
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
if basis is None and not replace_none:
|
||
|
|
return None
|
||
|
|
|
||
|
|
if basis is None:
|
||
|
|
if isinstance(expr, KetBase):
|
||
|
|
return _make_default(expr.__class__)
|
||
|
|
elif isinstance(expr, BraBase):
|
||
|
|
return _make_default(expr.dual_class())
|
||
|
|
elif isinstance(expr, Operator):
|
||
|
|
state_inst = operators_to_state(expr)
|
||
|
|
return (state_inst if state_inst is not None else None)
|
||
|
|
else:
|
||
|
|
return None
|
||
|
|
elif (isinstance(basis, Operator) or
|
||
|
|
(not isinstance(basis, StateBase) and issubclass(basis, Operator))):
|
||
|
|
state = operators_to_state(basis)
|
||
|
|
if state is None:
|
||
|
|
return None
|
||
|
|
elif isinstance(state, StateBase):
|
||
|
|
return state
|
||
|
|
else:
|
||
|
|
return _make_default(state)
|
||
|
|
elif isinstance(basis, StateBase):
|
||
|
|
return basis
|
||
|
|
elif issubclass(basis, StateBase):
|
||
|
|
return _make_default(basis)
|
||
|
|
else:
|
||
|
|
return None
|
||
|
|
|
||
|
|
|
||
|
|
def _make_default(expr):
|
||
|
|
# XXX: Catching TypeError like this is a bad way of distinguishing
|
||
|
|
# instances from classes. The logic using this function should be
|
||
|
|
# rewritten somehow.
|
||
|
|
try:
|
||
|
|
expr = expr()
|
||
|
|
except TypeError:
|
||
|
|
return expr
|
||
|
|
|
||
|
|
return expr
|
||
|
|
|
||
|
|
|
||
|
|
def enumerate_states(*args, **options):
|
||
|
|
"""
|
||
|
|
Returns instances of the given state with dummy indices appended
|
||
|
|
|
||
|
|
Operates in two different modes:
|
||
|
|
|
||
|
|
1. Two arguments are passed to it. The first is the base state which is to
|
||
|
|
be indexed, and the second argument is a list of indices to append.
|
||
|
|
|
||
|
|
2. Three arguments are passed. The first is again the base state to be
|
||
|
|
indexed. The second is the start index for counting. The final argument
|
||
|
|
is the number of kets you wish to receive.
|
||
|
|
|
||
|
|
Tries to call state._enumerate_state. If this fails, returns an empty list
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
args : list
|
||
|
|
See list of operation modes above for explanation
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.cartesian import XBra, XKet
|
||
|
|
>>> from sympy.physics.quantum.represent import enumerate_states
|
||
|
|
>>> test = XKet('foo')
|
||
|
|
>>> enumerate_states(test, 1, 3)
|
||
|
|
[|foo_1>, |foo_2>, |foo_3>]
|
||
|
|
>>> test2 = XBra('bar')
|
||
|
|
>>> enumerate_states(test2, [4, 5, 10])
|
||
|
|
[<bar_4|, <bar_5|, <bar_10|]
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
state = args[0]
|
||
|
|
|
||
|
|
if len(args) not in (2, 3):
|
||
|
|
raise NotImplementedError("Wrong number of arguments!")
|
||
|
|
|
||
|
|
if not isinstance(state, StateBase):
|
||
|
|
raise TypeError("First argument is not a state!")
|
||
|
|
|
||
|
|
if len(args) == 3:
|
||
|
|
num_states = args[2]
|
||
|
|
options['start_index'] = args[1]
|
||
|
|
else:
|
||
|
|
num_states = len(args[1])
|
||
|
|
options['index_list'] = args[1]
|
||
|
|
|
||
|
|
try:
|
||
|
|
ret = state._enumerate_state(num_states, **options)
|
||
|
|
except NotImplementedError:
|
||
|
|
ret = []
|
||
|
|
|
||
|
|
return ret
|