Vehicle-cpp/hkpc/software/sympy/physics/quantum/cartesian.py

"""Operators and states for 1D cartesian position and momentum.

TODO:

* Add 3D classes to mappings in operatorset.py

"""

from sympy.core.numbers import (I, pi)
from sympy.core.singleton import S
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.special.delta_functions import DiracDelta
from sympy.sets.sets import Interval

from sympy.physics.quantum.constants import hbar
from sympy.physics.quantum.hilbert import L2
from sympy.physics.quantum.operator import DifferentialOperator, HermitianOperator
from sympy.physics.quantum.state import Ket, Bra, State

__all__ = [
    'XOp',
    'YOp',
    'ZOp',
    'PxOp',
    'X',
    'Y',
    'Z',
    'Px',
    'XKet',
    'XBra',
    'PxKet',
    'PxBra',
    'PositionState3D',
    'PositionKet3D',
    'PositionBra3D'
]

#-------------------------------------------------------------------------
# Position operators
#-------------------------------------------------------------------------


class XOp(HermitianOperator):
    """1D cartesian position operator."""

    @classmethod
    def default_args(self):
        return ("X",)

    @classmethod
    def _eval_hilbert_space(self, args):
        return L2(Interval(S.NegativeInfinity, S.Infinity))

    def _eval_commutator_PxOp(self, other):
        return I*hbar

    def _apply_operator_XKet(self, ket, **options):
        return ket.position*ket

    def _apply_operator_PositionKet3D(self, ket, **options):
        return ket.position_x*ket

    def _represent_PxKet(self, basis, *, index=1, **options):
        states = basis._enumerate_state(2, start_index=index)
        coord1 = states[0].momentum
        coord2 = states[1].momentum
        d = DifferentialOperator(coord1)
        delta = DiracDelta(coord1 - coord2)

        return I*hbar*(d*delta)


class YOp(HermitianOperator):
    """ Y cartesian coordinate operator (for 2D or 3D systems) """

    @classmethod
    def default_args(self):
        return ("Y",)

    @classmethod
    def _eval_hilbert_space(self, args):
        return L2(Interval(S.NegativeInfinity, S.Infinity))

    def _apply_operator_PositionKet3D(self, ket, **options):
        return ket.position_y*ket


class ZOp(HermitianOperator):
    """ Z cartesian coordinate operator (for 3D systems) """

    @classmethod
    def default_args(self):
        return ("Z",)

    @classmethod
    def _eval_hilbert_space(self, args):
        return L2(Interval(S.NegativeInfinity, S.Infinity))

    def _apply_operator_PositionKet3D(self, ket, **options):
        return ket.position_z*ket

#-------------------------------------------------------------------------
# Momentum operators
#-------------------------------------------------------------------------


class PxOp(HermitianOperator):
    """1D cartesian momentum operator."""

    @classmethod
    def default_args(self):
        return ("Px",)

    @classmethod
    def _eval_hilbert_space(self, args):
        return L2(Interval(S.NegativeInfinity, S.Infinity))

    def _apply_operator_PxKet(self, ket, **options):
        return ket.momentum*ket

    def _represent_XKet(self, basis, *, index=1, **options):
        states = basis._enumerate_state(2, start_index=index)
        coord1 = states[0].position
        coord2 = states[1].position
        d = DifferentialOperator(coord1)
        delta = DiracDelta(coord1 - coord2)

        return -I*hbar*(d*delta)

X = XOp('X')
Y = YOp('Y')
Z = ZOp('Z')
Px = PxOp('Px')

#-------------------------------------------------------------------------
# Position eigenstates
#-------------------------------------------------------------------------


class XKet(Ket):
    """1D cartesian position eigenket."""

    @classmethod
    def _operators_to_state(self, op, **options):
        return self.__new__(self, *_lowercase_labels(op), **options)

    def _state_to_operators(self, op_class, **options):
        return op_class.__new__(op_class,
                                *_uppercase_labels(self), **options)

    @classmethod
    def default_args(self):
        return ("x",)

    @classmethod
    def dual_class(self):
        return XBra

    @property
    def position(self):
        """The position of the state."""
        return self.label[0]

    def _enumerate_state(self, num_states, **options):
        return _enumerate_continuous_1D(self, num_states, **options)

    def _eval_innerproduct_XBra(self, bra, **hints):
        return DiracDelta(self.position - bra.position)

    def _eval_innerproduct_PxBra(self, bra, **hints):
        return exp(-I*self.position*bra.momentum/hbar)/sqrt(2*pi*hbar)


class XBra(Bra):
    """1D cartesian position eigenbra."""

    @classmethod
    def default_args(self):
        return ("x",)

    @classmethod
    def dual_class(self):
        return XKet

    @property
    def position(self):
        """The position of the state."""
        return self.label[0]


class PositionState3D(State):
    """ Base class for 3D cartesian position eigenstates """

    @classmethod
    def _operators_to_state(self, op, **options):
        return self.__new__(self, *_lowercase_labels(op), **options)

    def _state_to_operators(self, op_class, **options):
        return op_class.__new__(op_class,
                                *_uppercase_labels(self), **options)

    @classmethod
    def default_args(self):
        return ("x", "y", "z")

    @property
    def position_x(self):
        """ The x coordinate of the state """
        return self.label[0]

    @property
    def position_y(self):
        """ The y coordinate of the state """
        return self.label[1]

    @property
    def position_z(self):
        """ The z coordinate of the state """
        return self.label[2]


class PositionKet3D(Ket, PositionState3D):
    """ 3D cartesian position eigenket """

    def _eval_innerproduct_PositionBra3D(self, bra, **options):
        x_diff = self.position_x - bra.position_x
        y_diff = self.position_y - bra.position_y
        z_diff = self.position_z - bra.position_z

        return DiracDelta(x_diff)*DiracDelta(y_diff)*DiracDelta(z_diff)

    @classmethod
    def dual_class(self):
        return PositionBra3D


# XXX: The type:ignore here is because mypy gives Definition of
# "_state_to_operators" in base class "PositionState3D" is incompatible with
# definition in base class "BraBase"
class PositionBra3D(Bra, PositionState3D):  # type: ignore
    """ 3D cartesian position eigenbra """

    @classmethod
    def dual_class(self):
        return PositionKet3D

#-------------------------------------------------------------------------
# Momentum eigenstates
#-------------------------------------------------------------------------


class PxKet(Ket):
    """1D cartesian momentum eigenket."""

    @classmethod
    def _operators_to_state(self, op, **options):
        return self.__new__(self, *_lowercase_labels(op), **options)

    def _state_to_operators(self, op_class, **options):
        return op_class.__new__(op_class,
                                *_uppercase_labels(self), **options)

    @classmethod
    def default_args(self):
        return ("px",)

    @classmethod
    def dual_class(self):
        return PxBra

    @property
    def momentum(self):
        """The momentum of the state."""
        return self.label[0]

    def _enumerate_state(self, *args, **options):
        return _enumerate_continuous_1D(self, *args, **options)

    def _eval_innerproduct_XBra(self, bra, **hints):
        return exp(I*self.momentum*bra.position/hbar)/sqrt(2*pi*hbar)

    def _eval_innerproduct_PxBra(self, bra, **hints):
        return DiracDelta(self.momentum - bra.momentum)


class PxBra(Bra):
    """1D cartesian momentum eigenbra."""

    @classmethod
    def default_args(self):
        return ("px",)

    @classmethod
    def dual_class(self):
        return PxKet

    @property
    def momentum(self):
        """The momentum of the state."""
        return self.label[0]

#-------------------------------------------------------------------------
# Global helper functions
#-------------------------------------------------------------------------


def _enumerate_continuous_1D(*args, **options):
    state = args[0]
    num_states = args[1]
    state_class = state.__class__
    index_list = options.pop('index_list', [])

    if len(index_list) == 0:
        start_index = options.pop('start_index', 1)
        index_list = list(range(start_index, start_index + num_states))

    enum_states = [0 for i in range(len(index_list))]

    for i, ind in enumerate(index_list):
        label = state.args[0]
        enum_states[i] = state_class(str(label) + "_" + str(ind), **options)

    return enum_states


def _lowercase_labels(ops):
    if not isinstance(ops, set):
        ops = [ops]

    return [str(arg.label[0]).lower() for arg in ops]


def _uppercase_labels(ops):
    if not isinstance(ops, set):
        ops = [ops]

    new_args = [str(arg.label[0])[0].upper() +
                str(arg.label[0])[1:] for arg in ops]

    return new_args