651 lines
19 KiB
Python
651 lines
19 KiB
Python
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"""Quantum mechanical operators.
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TODO:
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* Fix early 0 in apply_operators.
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* Debug and test apply_operators.
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* Get cse working with classes in this file.
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* Doctests and documentation of special methods for InnerProduct, Commutator,
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AntiCommutator, represent, apply_operators.
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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.function import (Derivative, expand)
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from sympy.core.mul import Mul
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from sympy.core.numbers import oo
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from sympy.core.singleton import S
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from sympy.printing.pretty.stringpict import prettyForm
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from sympy.physics.quantum.dagger import Dagger
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from sympy.physics.quantum.qexpr import QExpr, dispatch_method
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from sympy.matrices import eye
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__all__ = [
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'Operator',
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'HermitianOperator',
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'UnitaryOperator',
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'IdentityOperator',
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'OuterProduct',
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'DifferentialOperator'
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]
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#-----------------------------------------------------------------------------
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# Operators and outer products
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#-----------------------------------------------------------------------------
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class Operator(QExpr):
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"""Base class for non-commuting quantum operators.
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An operator maps between quantum states [1]_. In quantum mechanics,
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observables (including, but not limited to, measured physical values) are
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represented as Hermitian operators [2]_.
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Parameters
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==========
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args : tuple
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The list of numbers or parameters that uniquely specify the
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operator. For time-dependent operators, this will include the time.
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Examples
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========
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Create an operator and examine its attributes::
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>>> from sympy.physics.quantum import Operator
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>>> from sympy import I
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>>> A = Operator('A')
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>>> A
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A
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>>> A.hilbert_space
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H
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>>> A.label
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(A,)
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>>> A.is_commutative
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False
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Create another operator and do some arithmetic operations::
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>>> B = Operator('B')
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>>> C = 2*A*A + I*B
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>>> C
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2*A**2 + I*B
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Operators do not commute::
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>>> A.is_commutative
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False
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>>> B.is_commutative
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False
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>>> A*B == B*A
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False
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Polymonials of operators respect the commutation properties::
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>>> e = (A+B)**3
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>>> e.expand()
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A*B*A + A*B**2 + A**2*B + A**3 + B*A*B + B*A**2 + B**2*A + B**3
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Operator inverses are handle symbolically::
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>>> A.inv()
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A**(-1)
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>>> A*A.inv()
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1
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Operator_%28physics%29
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.. [2] https://en.wikipedia.org/wiki/Observable
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"""
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@classmethod
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def default_args(self):
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return ("O",)
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#-------------------------------------------------------------------------
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# Printing
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#-------------------------------------------------------------------------
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_label_separator = ','
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def _print_operator_name(self, printer, *args):
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return self.__class__.__name__
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_print_operator_name_latex = _print_operator_name
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def _print_operator_name_pretty(self, printer, *args):
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return prettyForm(self.__class__.__name__)
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def _print_contents(self, printer, *args):
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if len(self.label) == 1:
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return self._print_label(printer, *args)
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else:
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return '%s(%s)' % (
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self._print_operator_name(printer, *args),
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self._print_label(printer, *args)
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)
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def _print_contents_pretty(self, printer, *args):
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if len(self.label) == 1:
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return self._print_label_pretty(printer, *args)
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else:
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pform = self._print_operator_name_pretty(printer, *args)
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label_pform = self._print_label_pretty(printer, *args)
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label_pform = prettyForm(
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*label_pform.parens(left='(', right=')')
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)
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pform = prettyForm(*pform.right(label_pform))
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return pform
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def _print_contents_latex(self, printer, *args):
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if len(self.label) == 1:
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return self._print_label_latex(printer, *args)
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else:
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return r'%s\left(%s\right)' % (
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self._print_operator_name_latex(printer, *args),
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self._print_label_latex(printer, *args)
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)
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#-------------------------------------------------------------------------
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# _eval_* methods
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#-------------------------------------------------------------------------
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def _eval_commutator(self, other, **options):
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"""Evaluate [self, other] if known, return None if not known."""
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return dispatch_method(self, '_eval_commutator', other, **options)
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def _eval_anticommutator(self, other, **options):
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"""Evaluate [self, other] if known."""
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return dispatch_method(self, '_eval_anticommutator', other, **options)
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#-------------------------------------------------------------------------
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# Operator application
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#-------------------------------------------------------------------------
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def _apply_operator(self, ket, **options):
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return dispatch_method(self, '_apply_operator', ket, **options)
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def matrix_element(self, *args):
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raise NotImplementedError('matrix_elements is not defined')
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def inverse(self):
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return self._eval_inverse()
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inv = inverse
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def _eval_inverse(self):
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return self**(-1)
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def __mul__(self, other):
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if isinstance(other, IdentityOperator):
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return self
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return Mul(self, other)
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class HermitianOperator(Operator):
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"""A Hermitian operator that satisfies H == Dagger(H).
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Parameters
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==========
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args : tuple
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The list of numbers or parameters that uniquely specify the
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operator. For time-dependent operators, this will include the time.
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Examples
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========
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>>> from sympy.physics.quantum import Dagger, HermitianOperator
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>>> H = HermitianOperator('H')
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>>> Dagger(H)
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H
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"""
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is_hermitian = True
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def _eval_inverse(self):
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if isinstance(self, UnitaryOperator):
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return self
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else:
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return Operator._eval_inverse(self)
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def _eval_power(self, exp):
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if isinstance(self, UnitaryOperator):
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# so all eigenvalues of self are 1 or -1
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if exp.is_even:
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from sympy.core.singleton import S
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return S.One # is identity, see Issue 24153.
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elif exp.is_odd:
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return self
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# No simplification in all other cases
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return Operator._eval_power(self, exp)
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class UnitaryOperator(Operator):
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"""A unitary operator that satisfies U*Dagger(U) == 1.
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Parameters
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==========
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args : tuple
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The list of numbers or parameters that uniquely specify the
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operator. For time-dependent operators, this will include the time.
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Examples
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========
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>>> from sympy.physics.quantum import Dagger, UnitaryOperator
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>>> U = UnitaryOperator('U')
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>>> U*Dagger(U)
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1
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"""
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def _eval_adjoint(self):
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return self._eval_inverse()
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class IdentityOperator(Operator):
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"""An identity operator I that satisfies op * I == I * op == op for any
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operator op.
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Parameters
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==========
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N : Integer
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Optional parameter that specifies the dimension of the Hilbert space
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of operator. This is used when generating a matrix representation.
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Examples
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========
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>>> from sympy.physics.quantum import IdentityOperator
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>>> IdentityOperator()
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I
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"""
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@property
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def dimension(self):
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return self.N
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@classmethod
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def default_args(self):
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return (oo,)
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def __init__(self, *args, **hints):
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if not len(args) in (0, 1):
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raise ValueError('0 or 1 parameters expected, got %s' % args)
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self.N = args[0] if (len(args) == 1 and args[0]) else oo
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def _eval_commutator(self, other, **hints):
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return S.Zero
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def _eval_anticommutator(self, other, **hints):
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return 2 * other
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def _eval_inverse(self):
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return self
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def _eval_adjoint(self):
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return self
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def _apply_operator(self, ket, **options):
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return ket
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def _apply_from_right_to(self, bra, **options):
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return bra
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def _eval_power(self, exp):
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return self
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def _print_contents(self, printer, *args):
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return 'I'
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def _print_contents_pretty(self, printer, *args):
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return prettyForm('I')
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def _print_contents_latex(self, printer, *args):
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return r'{\mathcal{I}}'
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def __mul__(self, other):
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if isinstance(other, (Operator, Dagger)):
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return other
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return Mul(self, other)
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def _represent_default_basis(self, **options):
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if not self.N or self.N == oo:
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raise NotImplementedError('Cannot represent infinite dimensional' +
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' identity operator as a matrix')
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format = options.get('format', 'sympy')
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if format != 'sympy':
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raise NotImplementedError('Representation in format ' +
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'%s not implemented.' % format)
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return eye(self.N)
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class OuterProduct(Operator):
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"""An unevaluated outer product between a ket and bra.
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This constructs an outer product between any subclass of ``KetBase`` and
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``BraBase`` as ``|a><b|``. An ``OuterProduct`` inherits from Operator as they act as
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operators in quantum expressions. For reference see [1]_.
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Parameters
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==========
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ket : KetBase
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The ket on the left side of the outer product.
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bar : BraBase
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The bra on the right side of the outer product.
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Examples
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========
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Create a simple outer product by hand and take its dagger::
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>>> from sympy.physics.quantum import Ket, Bra, OuterProduct, Dagger
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>>> from sympy.physics.quantum import Operator
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>>> k = Ket('k')
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>>> b = Bra('b')
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>>> op = OuterProduct(k, b)
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>>> op
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|k><b|
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>>> op.hilbert_space
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H
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>>> op.ket
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|k>
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>>> op.bra
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<b|
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>>> Dagger(op)
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|b><k|
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In simple products of kets and bras outer products will be automatically
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identified and created::
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>>> k*b
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|k><b|
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But in more complex expressions, outer products are not automatically
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created::
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>>> A = Operator('A')
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>>> A*k*b
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A*|k>*<b|
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A user can force the creation of an outer product in a complex expression
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by using parentheses to group the ket and bra::
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>>> A*(k*b)
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A*|k><b|
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Outer_product
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"""
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is_commutative = False
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def __new__(cls, *args, **old_assumptions):
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from sympy.physics.quantum.state import KetBase, BraBase
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if len(args) != 2:
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raise ValueError('2 parameters expected, got %d' % len(args))
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ket_expr = expand(args[0])
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bra_expr = expand(args[1])
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if (isinstance(ket_expr, (KetBase, Mul)) and
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isinstance(bra_expr, (BraBase, Mul))):
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ket_c, kets = ket_expr.args_cnc()
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bra_c, bras = bra_expr.args_cnc()
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if len(kets) != 1 or not isinstance(kets[0], KetBase):
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raise TypeError('KetBase subclass expected'
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', got: %r' % Mul(*kets))
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if len(bras) != 1 or not isinstance(bras[0], BraBase):
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raise TypeError('BraBase subclass expected'
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', got: %r' % Mul(*bras))
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if not kets[0].dual_class() == bras[0].__class__:
|
||
|
|
raise TypeError(
|
||
|
|
'ket and bra are not dual classes: %r, %r' %
|
||
|
|
(kets[0].__class__, bras[0].__class__)
|
||
|
|
)
|
||
|
|
|
||
|
|
# TODO: make sure the hilbert spaces of the bra and ket are
|
||
|
|
# compatible
|
||
|
|
obj = Expr.__new__(cls, *(kets[0], bras[0]), **old_assumptions)
|
||
|
|
obj.hilbert_space = kets[0].hilbert_space
|
||
|
|
return Mul(*(ket_c + bra_c)) * obj
|
||
|
|
|
||
|
|
op_terms = []
|
||
|
|
if isinstance(ket_expr, Add) and isinstance(bra_expr, Add):
|
||
|
|
for ket_term in ket_expr.args:
|
||
|
|
for bra_term in bra_expr.args:
|
||
|
|
op_terms.append(OuterProduct(ket_term, bra_term,
|
||
|
|
**old_assumptions))
|
||
|
|
elif isinstance(ket_expr, Add):
|
||
|
|
for ket_term in ket_expr.args:
|
||
|
|
op_terms.append(OuterProduct(ket_term, bra_expr,
|
||
|
|
**old_assumptions))
|
||
|
|
elif isinstance(bra_expr, Add):
|
||
|
|
for bra_term in bra_expr.args:
|
||
|
|
op_terms.append(OuterProduct(ket_expr, bra_term,
|
||
|
|
**old_assumptions))
|
||
|
|
else:
|
||
|
|
raise TypeError(
|
||
|
|
'Expected ket and bra expression, got: %r, %r' %
|
||
|
|
(ket_expr, bra_expr)
|
||
|
|
)
|
||
|
|
|
||
|
|
return Add(*op_terms)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def ket(self):
|
||
|
|
"""Return the ket on the left side of the outer product."""
|
||
|
|
return self.args[0]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def bra(self):
|
||
|
|
"""Return the bra on the right side of the outer product."""
|
||
|
|
return self.args[1]
|
||
|
|
|
||
|
|
def _eval_adjoint(self):
|
||
|
|
return OuterProduct(Dagger(self.bra), Dagger(self.ket))
|
||
|
|
|
||
|
|
def _sympystr(self, printer, *args):
|
||
|
|
return printer._print(self.ket) + printer._print(self.bra)
|
||
|
|
|
||
|
|
def _sympyrepr(self, printer, *args):
|
||
|
|
return '%s(%s,%s)' % (self.__class__.__name__,
|
||
|
|
printer._print(self.ket, *args), printer._print(self.bra, *args))
|
||
|
|
|
||
|
|
def _pretty(self, printer, *args):
|
||
|
|
pform = self.ket._pretty(printer, *args)
|
||
|
|
return prettyForm(*pform.right(self.bra._pretty(printer, *args)))
|
||
|
|
|
||
|
|
def _latex(self, printer, *args):
|
||
|
|
k = printer._print(self.ket, *args)
|
||
|
|
b = printer._print(self.bra, *args)
|
||
|
|
return k + b
|
||
|
|
|
||
|
|
def _represent(self, **options):
|
||
|
|
k = self.ket._represent(**options)
|
||
|
|
b = self.bra._represent(**options)
|
||
|
|
return k*b
|
||
|
|
|
||
|
|
def _eval_trace(self, **kwargs):
|
||
|
|
# TODO if operands are tensorproducts this may be will be handled
|
||
|
|
# differently.
|
||
|
|
|
||
|
|
return self.ket._eval_trace(self.bra, **kwargs)
|
||
|
|
|
||
|
|
|
||
|
|
class DifferentialOperator(Operator):
|
||
|
|
"""An operator for representing the differential operator, i.e. d/dx
|
||
|
|
|
||
|
|
It is initialized by passing two arguments. The first is an arbitrary
|
||
|
|
expression that involves a function, such as ``Derivative(f(x), x)``. The
|
||
|
|
second is the function (e.g. ``f(x)``) which we are to replace with the
|
||
|
|
``Wavefunction`` that this ``DifferentialOperator`` is applied to.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
expr : Expr
|
||
|
|
The arbitrary expression which the appropriate Wavefunction is to be
|
||
|
|
substituted into
|
||
|
|
|
||
|
|
func : Expr
|
||
|
|
A function (e.g. f(x)) which is to be replaced with the appropriate
|
||
|
|
Wavefunction when this DifferentialOperator is applied
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
You can define a completely arbitrary expression and specify where the
|
||
|
|
Wavefunction is to be substituted
|
||
|
|
|
||
|
|
>>> from sympy import Derivative, Function, Symbol
|
||
|
|
>>> from sympy.physics.quantum.operator import DifferentialOperator
|
||
|
|
>>> from sympy.physics.quantum.state import Wavefunction
|
||
|
|
>>> from sympy.physics.quantum.qapply import qapply
|
||
|
|
>>> f = Function('f')
|
||
|
|
>>> x = Symbol('x')
|
||
|
|
>>> d = DifferentialOperator(1/x*Derivative(f(x), x), f(x))
|
||
|
|
>>> w = Wavefunction(x**2, x)
|
||
|
|
>>> d.function
|
||
|
|
f(x)
|
||
|
|
>>> d.variables
|
||
|
|
(x,)
|
||
|
|
>>> qapply(d*w)
|
||
|
|
Wavefunction(2, x)
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
@property
|
||
|
|
def variables(self):
|
||
|
|
"""
|
||
|
|
Returns the variables with which the function in the specified
|
||
|
|
arbitrary expression is evaluated
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.operator import DifferentialOperator
|
||
|
|
>>> from sympy import Symbol, Function, Derivative
|
||
|
|
>>> x = Symbol('x')
|
||
|
|
>>> f = Function('f')
|
||
|
|
>>> d = DifferentialOperator(1/x*Derivative(f(x), x), f(x))
|
||
|
|
>>> d.variables
|
||
|
|
(x,)
|
||
|
|
>>> y = Symbol('y')
|
||
|
|
>>> d = DifferentialOperator(Derivative(f(x, y), x) +
|
||
|
|
... Derivative(f(x, y), y), f(x, y))
|
||
|
|
>>> d.variables
|
||
|
|
(x, y)
|
||
|
|
"""
|
||
|
|
|
||
|
|
return self.args[-1].args
|
||
|
|
|
||
|
|
@property
|
||
|
|
def function(self):
|
||
|
|
"""
|
||
|
|
Returns the function which is to be replaced with the Wavefunction
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.operator import DifferentialOperator
|
||
|
|
>>> from sympy import Function, Symbol, Derivative
|
||
|
|
>>> x = Symbol('x')
|
||
|
|
>>> f = Function('f')
|
||
|
|
>>> d = DifferentialOperator(Derivative(f(x), x), f(x))
|
||
|
|
>>> d.function
|
||
|
|
f(x)
|
||
|
|
>>> y = Symbol('y')
|
||
|
|
>>> d = DifferentialOperator(Derivative(f(x, y), x) +
|
||
|
|
... Derivative(f(x, y), y), f(x, y))
|
||
|
|
>>> d.function
|
||
|
|
f(x, y)
|
||
|
|
"""
|
||
|
|
|
||
|
|
return self.args[-1]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def expr(self):
|
||
|
|
"""
|
||
|
|
Returns the arbitrary expression which is to have the Wavefunction
|
||
|
|
substituted into it
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.operator import DifferentialOperator
|
||
|
|
>>> from sympy import Function, Symbol, Derivative
|
||
|
|
>>> x = Symbol('x')
|
||
|
|
>>> f = Function('f')
|
||
|
|
>>> d = DifferentialOperator(Derivative(f(x), x), f(x))
|
||
|
|
>>> d.expr
|
||
|
|
Derivative(f(x), x)
|
||
|
|
>>> y = Symbol('y')
|
||
|
|
>>> d = DifferentialOperator(Derivative(f(x, y), x) +
|
||
|
|
... Derivative(f(x, y), y), f(x, y))
|
||
|
|
>>> d.expr
|
||
|
|
Derivative(f(x, y), x) + Derivative(f(x, y), y)
|
||
|
|
"""
|
||
|
|
|
||
|
|
return self.args[0]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def free_symbols(self):
|
||
|
|
"""
|
||
|
|
Return the free symbols of the expression.
|
||
|
|
"""
|
||
|
|
|
||
|
|
return self.expr.free_symbols
|
||
|
|
|
||
|
|
def _apply_operator_Wavefunction(self, func, **options):
|
||
|
|
from sympy.physics.quantum.state import Wavefunction
|
||
|
|
var = self.variables
|
||
|
|
wf_vars = func.args[1:]
|
||
|
|
|
||
|
|
f = self.function
|
||
|
|
new_expr = self.expr.subs(f, func(*var))
|
||
|
|
new_expr = new_expr.doit()
|
||
|
|
|
||
|
|
return Wavefunction(new_expr, *wf_vars)
|
||
|
|
|
||
|
|
def _eval_derivative(self, symbol):
|
||
|
|
new_expr = Derivative(self.expr, symbol)
|
||
|
|
return DifferentialOperator(new_expr, self.args[-1])
|
||
|
|
|
||
|
|
#-------------------------------------------------------------------------
|
||
|
|
# Printing
|
||
|
|
#-------------------------------------------------------------------------
|
||
|
|
|
||
|
|
def _print(self, printer, *args):
|
||
|
|
return '%s(%s)' % (
|
||
|
|
self._print_operator_name(printer, *args),
|
||
|
|
self._print_label(printer, *args)
|
||
|
|
)
|
||
|
|
|
||
|
|
def _print_pretty(self, printer, *args):
|
||
|
|
pform = self._print_operator_name_pretty(printer, *args)
|
||
|
|
label_pform = self._print_label_pretty(printer, *args)
|
||
|
|
label_pform = prettyForm(
|
||
|
|
*label_pform.parens(left='(', right=')')
|
||
|
|
)
|
||
|
|
pform = prettyForm(*pform.right(label_pform))
|
||
|
|
return pform
|