ó ¡¼™\c@ s¦dZddlmZmZddlmZmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZddlmZmZddlmZddlmZmZmZmZdd d d d gZd efd „ƒYZdefd„ƒYZd efd„ƒYZ d efd„ƒYZ!d„Z"d„Z#d„Z$d„Z%d„Z&d„Z'dd„Z)d„Z*d„Z+d„Z,d„Z-d„Z.dS(sClebsch-Gordon Coefficients.iÿÿÿÿ(tprint_functiontdivision( tAddtexpandtEqtExprtMult PiecewisetPowtsqrttSumtsymbolstsympifytWild(trange(t prettyFormt stringPict(tKroneckerDelta(tclebsch_gordant wigner_3jt wigner_6jt wigner_9jtCGtWigner3jtWigner6jtWigner9jtcg_simpcB s¡eZdZeZd„Zed„ƒZed„ƒZed„ƒZ ed„ƒZ ed„ƒZ ed„ƒZ ed„ƒZ d „Zd „Zd „ZRS( s‚Class for the Wigner-3j symbols Wigner 3j-symbols are coefficients determined by the coupling of two angular momenta. When created, they are expressed as symbolic quantities that, for numerical parameters, can be evaluated using the ``.doit()`` method [1]_. Parameters ========== j1, m1, j2, m2, j3, m3 : Number, Symbol Terms determining the angular momentum of coupled angular momentum systems. Examples ======== Declare a Wigner-3j coefficient and calculate its value >>> from sympy.physics.quantum.cg import Wigner3j >>> w3j = Wigner3j(6,0,4,0,2,0) >>> w3j Wigner3j(6, 0, 4, 0, 2, 0) >>> w3j.doit() sqrt(715)/143 See Also ======== CG: Clebsch-Gordan coefficients References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. cC s1tt||||||fƒ}tj||ŒS(N(tmapR Rt__new__(tclstj1tm1tj2tm2tj3tm3targs((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRFs!cC s |jdS(Ni(R$(tself((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRJscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRNscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR RscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR!VscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR"ZscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR#^scC s$tg|jD]}|j^q ƒ S(N(tallR$t is_number(R%targ((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyt is_symbolicbscG s'|j|jƒ|j|jƒf|j|jƒ|j|jƒf|j|jƒ|j|jƒff}d}d}dgd}xKtdƒD]=}tgtdƒD]}|||j ƒ^q¥ƒ||!!' ;    cG sDt|j|j|j|j|j|j|jfƒ}dt|ƒS(NsH\left(\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right)( RR+RR R"RR!R#ttuple(R%R3R$tlabel((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyt_latexŠscK sC|jrtdƒ‚nt|j|j|j|j|j|jƒS(NsCoefficients must be numerical( R)t ValueErrorRRR R"RR!R#(R%thints((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pytdoits (t__name__t __module__t__doc__tTruetis_commutativeRtpropertyRRR R!R"R#R)RARDRG(((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRs$  # cB s)eZdZd„Zd„Zd„ZRS(s<Class for Clebsch-Gordan coefficient Clebsch-Gordan coefficients describe the angular momentum coupling between two systems. The coefficients give the expansion of a coupled total angular momentum state and an uncoupled tensor product state. The Clebsch-Gordan coefficients are defined as [1]_: .. math :: C^{j_1,m_1}_{j_2,m_2,j_3,m_3} = \left\langle j_1,m_1;j_2,m_2 | j_3,m_3\right\rangle Parameters ========== j1, m1, j2, m2, j3, m3 : Number, Symbol Terms determining the angular momentum of coupled angular momentum systems. Examples ======== Define a Clebsch-Gordan coefficient and evaluate its value >>> from sympy.physics.quantum.cg import CG >>> from sympy import S >>> cg = CG(S(3)/2, S(3)/2, S(1)/2, -S(1)/2, 1, 1) >>> cg CG(3/2, 3/2, 1/2, -1/2, 1, 1) >>> cg.doit() sqrt(3)/2 See Also ======== Wigner3j: Wigner-3j symbols References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. cK sC|jrtdƒ‚nt|j|j|j|j|j|jƒS(NsCoefficients must be numerical( R)RERRR R"RR!R#(R%RF((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRGÀs cG sE|j|j|j|j|jfddƒ}|j|j|jfddƒ}t|jƒ|jƒƒ}t |j dƒŒ}t |j dƒŒ}||jƒksËt |j d||jƒƒŒ}n||jƒkst |j d||jƒƒŒ}nt dd|ƒ}t |j |ƒŒ}t |j|ƒŒ}|S(Nt delimitert,R*tC(t _print_seqRRR R!R"R#R,R-RR0R/RR1tabove(R%R3R$tbotttoptpadR<((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRAÅs'!&&cG sDt|j|j|j|j|j|j|jfƒ}dt|ƒS(NsC^{%s,%s}_{%s,%s,%s,%s}( RR+R"R#RRR R!RB(R%R3R$RC((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRD×s(RHRIRJRGRARD(((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR–s(  cB s›eZdZd„Zed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZ ed„ƒZ ed„ƒZ d „Z d „Z d „ZRS( saClass for the Wigner-6j symbols See Also ======== Wigner3j: Wigner-3j symbols cC s1tt||||||fƒ}tj||ŒS(N(RR RR(RRR tj12R"R8tj23R$((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRæs!cC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRêscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR îscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRVòscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR"öscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR8úscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRWþscC s$tg|jD]}|j^q ƒ S(N(R&R$R'(R%R(((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR)scG s3|j|jƒ|j|jƒf|j|jƒ|j|jƒf|j|jƒ|j|jƒff}d}d}dgd}xKtdƒD]=}tgtdƒD]}|||j ƒ^q¥ƒ||R?R@((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRAs>!!' ;    cG sDt|j|j|j|j|j|j|jfƒ}dt|ƒS(NsJ\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right\}( RR+RR RVR"R8RWRB(R%R3R$RC((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRD*scK sC|jrtdƒ‚nt|j|j|j|j|j|jƒS(NsCoefficients must be numerical( R)RERRR RVR"R8RW(R%RF((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRG0s (RHRIRJRRMRR RVR"R8RWR)RARDRG(((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRÝs  # cB sÈeZdZd„Zed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZ ed„ƒZ ed„ƒZ ed „ƒZ ed „ƒZ ed „ƒZd „Zd „Zd„ZRS(saClass for the Wigner-9j symbols See Also ======== Wigner3j: Wigner-3j symbols c C s:tt||||||||| f ƒ} tj|| ŒS(N(RR RR( RRR RVR"tj4tj34tj13tj24R8R$((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR?s*cC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRCscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR GscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRVKscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR"OscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRZSscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR[WscC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR\[scC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR]_scC s |jdS(Ni(R$(R%((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR8cscC s$tg|jD]}|j^q ƒ S(N(R&R$R'(R%R(((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR)gscG s`|j|jƒ|j|jƒ|j|jƒf|j|jƒ|j|jƒ|j|jƒf|j|jƒ|j|jƒ|j|j ƒff}d}d}dgd}xKt dƒD]=}t gt dƒD]}|||j ƒ^qÒƒ||R?R@((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRAlsB**6 ;    c G sVt|j|j|j|j|j|j|j|j|j |j f ƒ}dt |ƒS(NsZ\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \\ %s & %s & %s \end{array}\right\}( RR+RR RVR"RZR[R\R]R8RB(R%R3R$RC((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRD’s!'c K sU|jrtdƒ‚nt|j|j|j|j|j|j|j |j |j ƒ S(NsCoefficients must be numerical( R)RERRR RVR"RZR[R\R]R8(R%RF((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRG˜s (RHRIRJRRMRR RVR"RZR[R\R]R8R)RARDRG(((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyR6s  & cC s—t|tƒrt|ƒSt|tƒr2t|ƒSt|tƒrgtg|jD]}t|ƒ^qNŒSt|tƒrtt|j ƒ|j ƒS|SdS(s¥Simplify and combine CG coefficients This function uses various symmetry and properties of sums and products of Clebsch-Gordan coefficients to simplify statements involving these terms [1]_. Examples ======== Simplify the sum over CG(a,alpha,0,0,a,alpha) for all alpha to 2*a+1 >>> from sympy.physics.quantum.cg import CG, cg_simp >>> a = CG(1,1,0,0,1,1) >>> b = CG(1,0,0,0,1,0) >>> c = CG(1,-1,0,0,1,-1) >>> cg_simp(a+b+c) 3 See Also ======== CG: Clebsh-Gordan coefficients References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. N( t isinstanceRt _cg_simp_addR t _cg_simp_sumRR$RRtbasetexp(teR(((s7lib/python2.7/site-packages/sympy/physics/quantum/cg.pyRžs  &cC srg}g}t|ƒ}xæ|jD]Û}|jtƒrðt|tƒr\|jt|ƒƒqýt|tƒràd}x=|jD]2}t|tƒr£|t|ƒ9}q{||9}q{W|jtƒrÐ|j|ƒqí|j|ƒqý|j|ƒq"|j|ƒq"Wt |ƒ\}}|j|ƒt |ƒ\}}|j|ƒt |ƒ\}}|j|ƒt |Œt |ŒS(sTakes a sum of terms involving Clebsch-Gordan coefficients and simplifies the terms. First, we create two lists, cg_part, which is all the terms involving CG coefficients, and other_part, which is all other terms. 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