o
    *ήcJ                     @   s   d Z ddlmZ ddlmZ ddlmZmZ ddlm	Z	 ddl
mZ ddlmZ ddlmZ dd	lmZ dd
lmZmZ ddlmZ g dZG dd deZG dd deZG dd deZG dd deZG dd deZG dd deZdS )a  Quantum mechanical operators.

TODO:

* Fix early 0 in apply_operators.
* Debug and test apply_operators.
* Get cse working with classes in this file.
* Doctests and documentation of special methods for InnerProduct, Commutator,
  AntiCommutator, represent, apply_operators.
    )Add)Expr)
Derivativeexpand)Muloo)S
prettyForm)Dagger)QExprdispatch_method)eye)OperatorHermitianOperatorUnitaryOperatorIdentityOperatorOuterProductDifferentialOperatorc                   @   s   e Zd ZdZedd ZdZdd ZeZdd Z	d	d
 Z
dd Zdd Zdd Zdd Zdd Zdd Zdd ZeZdd Zdd ZdS )r   a
  Base class for non-commuting quantum operators.

    An operator maps between quantum states [1]_. In quantum mechanics,
    observables (including, but not limited to, measured physical values) are
    represented as Hermitian operators [2]_.

    Parameters
    ==========

    args : tuple
        The list of numbers or parameters that uniquely specify the
        operator. For time-dependent operators, this will include the time.

    Examples
    ========

    Create an operator and examine its attributes::

        >>> from sympy.physics.quantum import Operator
        >>> from sympy import I
        >>> A = Operator('A')
        >>> A
        A
        >>> A.hilbert_space
        H
        >>> A.label
        (A,)
        >>> A.is_commutative
        False

    Create another operator and do some arithmetic operations::

        >>> B = Operator('B')
        >>> C = 2*A*A + I*B
        >>> C
        2*A**2 + I*B

    Operators do not commute::

        >>> A.is_commutative
        False
        >>> B.is_commutative
        False
        >>> A*B == B*A
        False

    Polymonials of operators respect the commutation properties::

        >>> e = (A+B)**3
        >>> e.expand()
        A*B*A + A*B**2 + A**2*B + A**3 + B*A*B + B*A**2 + B**2*A + B**3

    Operator inverses are handle symbolically::

        >>> A.inv()
        A**(-1)
        >>> A*A.inv()
        1

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Operator_%28physics%29
    .. [2] https://en.wikipedia.org/wiki/Observable
    c                 C      dS )N)O selfr   r   E/tmp/pip-target-vg8gfxp4/lib/python/sympy/physics/quantum/operator.pydefault_argsh   s   zOperator.default_args,c                 G      | j jS N)	__class____name__r   printerargsr   r   r   _print_operator_namer      zOperator._print_operator_namec                 G   s   t | jjS r   )r   r    r!   r"   r   r   r   _print_operator_name_prettyw   s   z$Operator._print_operator_name_prettyc                 G   H   t | jdkr| j|g|R  S d| j|g|R  | j|g|R  f S )N   %s(%s))lenlabel_print_labelr%   r"   r   r   r   _print_contentsz      zOperator._print_contentsc                 G   sh   t | jdkr| j|g|R  S | j|g|R  }| j|g|R  }t|jddd }t|| }|S )Nr)   ()leftright)r+   r,   _print_label_prettyr'   r   parensr4   r   r#   r$   pformlabel_pformr   r   r   _print_contents_pretty   s   zOperator._print_contents_prettyc                 G   r(   )Nr)   z%s\left(%s\right))r+   r,   _print_label_latex_print_operator_name_latexr"   r   r   r   _print_contents_latex   r/   zOperator._print_contents_latexc                 K      t | d|fi |S )z:Evaluate [self, other] if known, return None if not known._eval_commutatorr   r   otheroptionsr   r   r   r?         zOperator._eval_commutatorc                 K   r>   )z Evaluate [self, other] if known._eval_anticommutatorr@   rA   r   r   r   rE      rD   zOperator._eval_anticommutatorc                 K   r>   )N_apply_operatorr@   r   ketrC   r   r   r   rF      s   zOperator._apply_operatorc                 G   s   t d)Nzmatrix_elements is not defined)NotImplementedError)r   r$   r   r   r   matrix_element   r&   zOperator.matrix_elementc                 C      |   S r   _eval_inverser   r   r   r   inverse   r&   zOperator.inversec                 C   s   | d S Nr   r   r   r   r   rM      r&   zOperator._eval_inversec                 C   s   t |tr| S t| |S r   )
isinstancer   r   r   rB   r   r   r   __mul__   s   

zOperator.__mul__N)r!   
__module____qualname____doc__classmethodr   _label_separatorr%   r<   r'   r.   r:   r=   r?   rE   rF   rJ   rN   invrM   rS   r   r   r   r   r   %   s&    B
	r   c                   @   s$   e Zd ZdZdZdd Zdd ZdS )r   a  A Hermitian operator that satisfies H == Dagger(H).

    Parameters
    ==========

    args : tuple
        The list of numbers or parameters that uniquely specify the
        operator. For time-dependent operators, this will include the time.

    Examples
    ========

    >>> from sympy.physics.quantum import Dagger, HermitianOperator
    >>> H = HermitianOperator('H')
    >>> Dagger(H)
    H
    Tc                 C   s   t | tr| S t| S r   )rQ   r   r   rM   r   r   r   r   rM      s   

zHermitianOperator._eval_inversec                 C   sL   t | tr |dkrt| |S t|d dkr| t|  S | S t| |S )NrP      r   )rQ   r   r   _eval_powerabsrM   r   expr   r   r   r[      s   
zHermitianOperator._eval_powerN)r!   rT   rU   rV   is_hermitianrM   r[   r   r   r   r   r      s
    r   c                   @   s   e Zd ZdZdd ZdS )r   a  A unitary operator that satisfies U*Dagger(U) == 1.

    Parameters
    ==========

    args : tuple
        The list of numbers or parameters that uniquely specify the
        operator. For time-dependent operators, this will include the time.

    Examples
    ========

    >>> from sympy.physics.quantum import Dagger, UnitaryOperator
    >>> U = UnitaryOperator('U')
    >>> U*Dagger(U)
    1
    c                 C   rK   r   rL   r   r   r   r   _eval_adjoint   r&   zUnitaryOperator._eval_adjointN)r!   rT   rU   rV   r`   r   r   r   r   r      s    r   c                   @   s   e Zd ZdZedd Zedd Zdd Zdd	 Z	d
d Z
dd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd ZdS )r   a  An identity operator I that satisfies op * I == I * op == op for any
    operator op.

    Parameters
    ==========

    N : Integer
        Optional parameter that specifies the dimension of the Hilbert space
        of operator. This is used when generating a matrix representation.

    Examples
    ========

    >>> from sympy.physics.quantum import IdentityOperator
    >>> IdentityOperator()
    I
    c                 C   s   | j S r   )Nr   r   r   r   	dimension     zIdentityOperator.dimensionc                 C   s   t fS r   r   r   r   r   r   r     rc   zIdentityOperator.default_argsc                 O   sD   t |dvrtd| t |dkr|d r|d | _d S t| _d S )N)r   r)   z"0 or 1 parameters expected, got %sr)   r   )r+   
ValueErrorr   ra   )r   r$   hintsr   r   r   __init__  s   ,zIdentityOperator.__init__c                 K   s   t jS r   )r	   Zeror   rB   re   r   r   r   r?     s   z!IdentityOperator._eval_commutatorc                 K   s   d| S )NrZ   r   rh   r   r   r   rE     r&   z%IdentityOperator._eval_anticommutatorc                 C      | S r   r   r   r   r   r   rM   "     zIdentityOperator._eval_inversec                 C   ri   r   r   r   r   r   r   r`   %  rj   zIdentityOperator._eval_adjointc                 K   s   |S r   r   rG   r   r   r   rF   (  rj   z IdentityOperator._apply_operatorc                 C   ri   r   r   r]   r   r   r   r[   +  rj   zIdentityOperator._eval_powerc                 G   r   NIr   r"   r   r   r   r.   .  rj   z IdentityOperator._print_contentsc                 G   s   t dS rk   r
   r"   r   r   r   r:   1  r&   z'IdentityOperator._print_contents_prettyc                 G   r   )Nz{\mathcal{I}}r   r"   r   r   r   r=   4  rj   z&IdentityOperator._print_contents_latexc                 C   s   t |ttfr	|S t| |S r   )rQ   r   r   r   rR   r   r   r   rS   7  s   
zIdentityOperator.__mul__c                 K   sF   | j r| j tkrtd|dd}|dkrtdd|  t| j S )NzCCannot represent infinite dimensional identity operator as a matrixformatsympyzRepresentation in format z%s not implemented.)ra   r   rI   getr   )r   rC   rm   r   r   r   _represent_default_basis>  s   
z)IdentityOperator._represent_default_basisN)r!   rT   rU   rV   propertyrb   rW   r   rf   r?   rE   rM   r`   rF   r[   r.   r:   r=   rS   rp   r   r   r   r   r      s$    

r   c                   @   sl   e Zd ZdZdZdd Zedd Zedd Zd	d
 Z	dd Z
dd Zdd Zdd Zdd Zdd ZdS )r   a  An unevaluated outer product between a ket and bra.

    This constructs an outer product between any subclass of ``KetBase`` and
    ``BraBase`` as ``|a><b|``. An ``OuterProduct`` inherits from Operator as they act as
    operators in quantum expressions.  For reference see [1]_.

    Parameters
    ==========

    ket : KetBase
        The ket on the left side of the outer product.
    bar : BraBase
        The bra on the right side of the outer product.

    Examples
    ========

    Create a simple outer product by hand and take its dagger::

        >>> from sympy.physics.quantum import Ket, Bra, OuterProduct, Dagger
        >>> from sympy.physics.quantum import Operator

        >>> k = Ket('k')
        >>> b = Bra('b')
        >>> op = OuterProduct(k, b)
        >>> op
        |k><b|
        >>> op.hilbert_space
        H
        >>> op.ket
        |k>
        >>> op.bra
        <b|
        >>> Dagger(op)
        |b><k|

    In simple products of kets and bras outer products will be automatically
    identified and created::

        >>> k*b
        |k><b|

    But in more complex expressions, outer products are not automatically
    created::

        >>> A = Operator('A')
        >>> A*k*b
        A*|k>*<b|

    A user can force the creation of an outer product in a complex expression
    by using parentheses to group the ket and bra::

        >>> A*(k*b)
        A*|k><b|

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Outer_product
    Fc                 O   s
  ddl m}m} t|dkrtdt| t|d }t|d }t||tfrt||tfr| \}}| \}	}
t|dksIt|d |sQt	dt|  t|
dks^t|
d |sft	dt|
  |d 
 |
d jkst	d|d j|
d jf tj| g|d |
d fR i |}|d j|_t||	  | S g }t|trt|tr|jD ]}|jD ]}|t||fi | qqt| S t|tr|jD ]}|t||fi | qt| S t|tr|jD ]}|t||fi | qt| S t	d	||f )
Nr   )KetBaseBraBaserZ   z2 parameters expected, got %dr)   z"KetBase subclass expected, got: %rz"BraBase subclass expected, got: %rz(ket and bra are not dual classes: %r, %rz,Expected ket and bra expression, got: %r, %r)sympy.physics.quantum.staterr   rs   r+   rd   r   rQ   r   args_cnc	TypeError
dual_classr    r   __new__hilbert_spacer   r$   appendr   )clsr$   old_assumptionsrr   rs   ket_exprbra_exprket_cketsbra_cbrasobjop_termsket_termbra_termr   r   r   rx     sj   $








zOuterProduct.__new__c                 C   
   | j d S )z5Return the ket on the left side of the outer product.r   r$   r   r   r   r   rH        
zOuterProduct.ketc                 C   r   )z6Return the bra on the right side of the outer product.r)   r   r   r   r   r   bra  r   zOuterProduct.brac                 C   s   t t| jt| jS r   )r   r   r   rH   r   r   r   r   r`     s   zOuterProduct._eval_adjointc                 G   s   | | j| | j S r   _printrH   r   r"   r   r   r   	_sympystr  s   zOuterProduct._sympystrc                 G   s2   d| j j|j| jg|R  |j| jg|R  f S )Nz	%s(%s,%s))r    r!   r   rH   r   r"   r   r   r   
_sympyrepr  s   $zOuterProduct._sympyreprc                 G   s2   | j j|g|R  }t|| jj|g|R   S r   )rH   _prettyr   r4   r   )r   r#   r$   r8   r   r   r   r     s   zOuterProduct._prettyc                 G   s0   |j | jg|R  }|j | jg|R  }|| S r   r   )r   r#   r$   kbr   r   r   _latex  s   zOuterProduct._latexc                 K   s,   | j jdi |}| jjdi |}|| S )Nr   )rH   
_representr   )r   rC   r   r   r   r   r   r     s   zOuterProduct._representc                 K   s   | j j| jfi |S r   )rH   _eval_tracer   )r   kwargsr   r   r   r     s   zOuterProduct._eval_traceN)r!   rT   rU   rV   is_commutativerx   rq   rH   r   r`   r   r   r   r   r   r   r   r   r   r   r   K  s    <8

r   c                   @   s`   e Zd ZdZedd Zedd Zedd Zedd	 Zd
d Z	dd Z
dd Zdd ZdS )r   a+  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)

    c                 C   s   | j d j S )a  
        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)
        rP   r   r   r   r   r   	variables  s   zDifferentialOperator.variablesc                 C   r   )ad  
        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)
        rP   r   r   r   r   r   function.  s   
zDifferentialOperator.functionc                 C   r   )a  
        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)
        r   r   r   r   r   r   exprF  s   
zDifferentialOperator.exprc                 C   r   )z<
        Return the free symbols of the expression.
        )r   free_symbolsr   r   r   r   r   _  s   z!DifferentialOperator.free_symbolsc                 C   sP   ddl m} | j}|jdd  }| j}| j||| }| }||g|R  S )Nr   )Wavefunctionr)   )rt   r   r   r$   r   r   subsdoit)r   funcr   varwf_varsfnew_exprr   r   r   _apply_operator_Wavefunctiong  s   z1DifferentialOperator._apply_operator_Wavefunctionc                 C   s   t | j|}t|| jd S rO   )r   r   r   r$   )r   symbolr   r   r   r   _eval_derivativer  s   z%DifferentialOperator._eval_derivativec                 G   s(   d| j |g|R  | j|g|R  f S )Nr*   )r%   r-   r"   r   r   r   r   z  s   zDifferentialOperator._printc                 G   sH   | j |g|R  }| j|g|R  }t|jddd }t|| }|S )Nr0   r1   r2   )r'   r5   r   r6   r4   r7   r   r   r   _print_pretty  s   z"DifferentialOperator._print_prettyN)r!   rT   rU   rV   rq   r   r   r   r   r   r   r   r   r   r   r   r   r     s    )



r   N)rV   sympy.core.addr   sympy.core.exprr   sympy.core.functionr   r   sympy.core.mulr   sympy.core.numbersr   sympy.core.singletonr	    sympy.printing.pretty.stringpictr   sympy.physics.quantum.daggerr   sympy.physics.quantum.qexprr   r   sympy.matricesr   __all__r   r   r   r   r   r   r   r   r   r   <module>   s(     'O !