ó ¡¼™\c@ s6dZddlmZmZddlmZmZmZmZm Z m Z m Z ddl m Z ddlmZddlmZmZddlmZdd d d d d gZdefd„ƒYZd efd„ƒYZd efd„ƒYZd efd„ƒYZd efd„ƒYZd efd„ƒYZdS(sQuantum 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. iÿÿÿÿ(tprint_functiontdivision(t DerivativetExprtIntegertootMultexpandtAdd(t prettyForm(tDagger(tQExprtdispatch_method(teyetOperatortHermitianOperatortUnitaryOperatortIdentityOperatort OuterProducttDifferentialOperatorcB s›eZdZed„ƒZdZd„ZeZd„Zd„Z d„Z d„Z d„Z d „Z d „Zd „Zd „ZeZd „Zd„ZRS(sBase 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 symbols, 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 don't 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 cC sdS(NtO(R((tself((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt default_argsest,cG s|j|jj|ŒS(N(t_printt __class__t__name__(Rtprintertargs((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt_print_operator_nameoscG st|jjƒS(N(R RR(RRR((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt_print_operator_name_prettytscG sOt|jƒdkr%|j||ŒSd|j||Œ|j||ŒfSdS(Nis%s(%s)(tlentlabelt _print_labelR(RRR((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt_print_contentsws cG s„t|jƒdkr%|j||ŒS|j||Œ}|j||Œ}t|jddddƒŒ}t|j|ƒŒ}|SdS(Nitleftt(trightt)(RR t_print_label_prettyRR tparensR%(RRRtpformt label_pform((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt_print_contents_pretty€scG sOt|jƒdkr%|j||ŒSd|j||Œ|j||ŒfSdS(Nis%s\left(%s\right)(RR t_print_label_latext_print_operator_name_latex(RRR((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt_print_contents_latexŒs cK st|d||S(s:Evaluate [self, other] if known, return None if not known.t_eval_commutator(R (Rtothertoptions((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR/™scK st|d||S(s Evaluate [self, other] if known.t_eval_anticommutator(R (RR0R1((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR2scK st|d||S(Nt_apply_operator(R (RtketR1((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR3¥scG stdƒ‚dS(Nsmatrix_elements is not defined(tNotImplementedError(RR((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pytmatrix_element¨scC s |jƒS(N(t _eval_inverse(R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pytinverse«scC s|dS(Niÿÿÿÿ((R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR7°scC s t|tƒr|St||ƒS(N(t isinstanceRR(RR0((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt__mul__³s(Rt __module__t__doc__t classmethodRt_label_separatorRR-RR"R+R.R/R2R3R6R8tinvR7R:(((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR"s"A        cB s&eZdZeZd„Zd„ZRS(s”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 cC s$t|tƒr|Stj|ƒSdS(N(R9RRR7(R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR7ÐscC smt|tƒrY|dkr+tj||ƒSt|ƒddkrR|tj|ƒS|Sntj||ƒSdS(Niÿÿÿÿii(R9RRt _eval_powertabsR7(Rtexp((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR@Ös (RR;R<tTruet is_hermitianR7R@(((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR»s cB seZdZd„ZRS(s’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 cC s |jƒS(N(R7(R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt _eval_adjointõs(RR;R<RE(((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyRâscB s˜eZdZed„ƒZed„ƒZd„Zd„Zd„Z d„Z d„Z d„Z d „Z d „Zd „Zd „Zd „Zd„ZRS(sšAn 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 cC s|jS(N(tN(R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt dimension scC stfS(N(R(R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyRscO sXt|ƒdkr%td|ƒ‚nt|ƒdkrK|drK|dnt|_dS(Niis"0 or 1 parameters expected, got %s(ii(Rt ValueErrorRRF(RRthints((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt__init__scK s tdƒS(Ni(R(RR0RI((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR/scK sd|S(Ni((RR0RI((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR2scC s|S(N((R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR7scC s|S(N((R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyRE"scK s|S(N((RR4R1((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR3%scC s|S(N((RRB((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR@(scG sdS(NtI((RRR((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR"+scG s tdƒS(NRK(R (RRR((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR+.scG sdS(Ns {\mathcal{I}}((RRR((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR.1scC s t|tƒr|St||ƒS(N(R9RR(RR0((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyR:4scK sn|j s|jtkr,tddƒ‚n|jddƒ}|dkratdd|ƒ‚nt|jƒS(Ns%Cannot represent infinite dimensionals identity operator as a matrixtformattsympysRepresentation in format s%s not implemented.(RFRR5tgetR (RR1RL((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt_represent_default_basis;s  (RR;R<tpropertyRGR=RRJR/R2R7RER3R@R"R+R.R:RO(((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyRùs           cB szeZdZeZd„Zed„ƒZed„ƒZd„Z d„Z d„Z d„Z d„Z d „Zd „ZRS( sÙAn unevaluated outer product between a ket and bra. This constructs an outer product between any subclass of ``KetBase`` and ``BraBase`` as ``|a>>> 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>>> op.hilbert_space H >>> op.ket |k> >>> op.bra >> Dagger(op) |b>>> k*b |k>>> A = Operator('A') >>> A*k*b A*|k>*>> A*(k*b) A*|k>>> 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) cC s|jdjS(s 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) iÿÿÿÿ(R(R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt variablesscC s |jdS(sd 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) iÿÿÿÿ(R(R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pytfunction+scC s |jdS(s¯ 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) i(R(R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pytexprCscC s |jjS(s< Return the free symbols of the expression. (Rut free_symbols(R((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyRv\scC scddlm}|j}|jd}|j}|jj|||Œƒ}|jƒ}|||ŒS(Niÿÿÿÿ(t Wavefunctioni(RSRwRsRRtRutsubstdoit(RtfuncRwtvartwf_varstftnew_expr((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt_apply_operator_Wavefunctionds    cC s&t|j|ƒ}t||jdƒS(Niÿÿÿÿ(RRuRR(RtsymbolR~((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt_eval_derivativeoscG s&d|j||Œ|j||ŒfS(Ns%s(%s)(RR!(RRR((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyRwscG s[|j||Œ}|j||Œ}t|jddddƒŒ}t|j|ƒŒ}|S(NR#R$R%R&(RR'R R(R%(RRRR)R*((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt _print_pretty}s ( RR;R<RPRsRtRuRvRRRR‚(((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyRès(  N(R<t __future__RRRMRRRRRRRt sympy.printing.pretty.stringpictR tsympy.physics.quantum.daggerR tsympy.physics.quantum.qexprR R tsympy.matricesR t__all__RRRRRR(((s=lib/python2.7/site-packages/sympy/physics/quantum/operator.pyt s$4 ™'O