+ i[Rt^RIHt^RIHt^RIHt^RIHt^RI H t ^RI H t ^RI Ht^RIHt^R IHtHt^R IHt^R IHt^R IHt^R IHtHt^RIHt^RIH t H!t!H"t"H#t#^RI$H%t%.R&Ot&!RR]4t'!RR]'4t(!RR]4t)!RR]4t*Rt+Rt,Rt-Rt.Rt/Rt0R'R lt1R!t2R"t3R#t4R$t5R%t6R#)(zClebsch-Gordon Coefficients.)Sum)Add)Expr)expand)Mul)Pow)Eq)S)Wildsymbols)sympify)sqrt) Piecewise) prettyForm stringPict)KroneckerDelta)clebsch_gordan wigner_3j wigner_6j wigner_9j) PRECEDENCECGWigner3jWigner6jWigner9jca]tRt^&toRtRtRt]R4t]R4t ]R4t ]R4t ]R4t ]R 4t ]R 4tR tR tR tRtVtR#)ra8Class for the Wigner-3j symbols. Explanation =========== 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. Tc X\\WW4WV34p\P!V.VO5!#Nmapr r__new__)clsj1m1j2m2j3m3argss&&&&&&& x/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/physics/quantum/cg.pyr Wigner3j.__new__Qs)7RRR45||C'$''c(VP^,#r(selfs&r)r" Wigner3j.j1Uyy|r+c(VP^,#r/r0s&r)r# Wigner3j.m1Yr3r+c(VP^,#r/r0s&r)r$ Wigner3j.j2]r3r+c(VP^,#r/r0s&r)r% Wigner3j.m2ar3r+c(VP^,#r/r0s&r)r& Wigner3j.j3er3r+c(VP^,#r/r0s&r)r' Wigner3j.m3ir3r+c\;QJd0RVP4F'dK R'*# R'*#!RVP44'*#)c38"TFqPxK R#5ir is_number.0args& r) 'Wigner3j.is_symbolic..o: }} FTallr(r0s&r) is_symbolicWigner3j.is_symbolicm<3: :33::3::3: ::::r+c^aaVPVP4VPVP43VPVP4VPVP43VPVP 4VPVP 433o^p^pR.^,p\^4F%o\VV3Rl\^444VS&K' Rp\^4EFpRp\^4FoSS,V,p VS,V P4, p V ^,p W, p \V PRV ,4!p \V PRV ,4!p VfT pK\VPRV,4!p\VPV 4!pK VfTpK\V4Fp \VPR4!pK \VPV4!pEK! \VP4!pV#)r:c3`<"TF#pSS,V,P4xK% R#5irwidthrMijms& r)rO#Wigner3j._pretty..z!<8a!A$q'--//8+.N )_printr"r#r$r%r&r'rangemaxr[rrightleftbelowparensr1printerr(hsepvsepmaxwDr]D_rowswdeltawleftwright_r^r_s&&* @@r)_prettyWigner3j._prettyrsnnTWW%w~~dgg'> ? ^^DGG $gnnTWW&= > ^^DGG $gnnTWW&= > @tAvqA<58< >$''477DGGTWWdggNNr+N)__name__ __module__ __qualname____firstlineno____doc__is_commutativer propertyr"r#r$r%r&r'rUrxr~r__static_attributes____classdictcell__ __classdict__s@r)rr&s&PN(;;!F OOr+cVa]tRt^toRt]R,^, tRtRtRt Rt Vt R#)ra_Class for Clebsch-Gordan coefficient. Explanation =========== Clebsch-Gordan coefficients describe the angular momentum coupling between two systems. The coefficients give the expansion of a coupled total angular momentum state and an uncoupled tensor product state. The Clebsch-Gordan coefficients are defined as [1]_: .. math :: C^{j_3,m_3}_{j_1,m_1,j_2,m_2} = \left\langle j_1,m_1;j_2,m_2 | j_3,m_3\right\rangle Parameters ========== j1, m1, j2, m2 : Number, Symbol Angular momenta of states 1 and 2. j3, m3: Number, Symbol Total angular momentum of the coupled system. Examples ======== Define a Clebsch-Gordan coefficient and evaluate its value >>> from sympy.physics.quantum.cg import CG >>> from sympy import S >>> cg = CG(S(3)/2, S(3)/2, S(1)/2, -S(1)/2, 1, 1) >>> cg CG(3/2, 3/2, 1/2, -1/2, 1, 1) >>> cg.doit() sqrt(3)/2 >>> CG(j1=S(1)/2, m1=-S(1)/2, j2=S(1)/2, m2=+S(1)/2, j3=1, m3=0).doit() sqrt(2)/2 Compare [2]_. See Also ======== Wigner3j: Wigner-3j symbols References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. .. [2] `Clebsch-Gordan Coefficients, Spherical Harmonics, and d Functions `_ in P.A. Zyla *et al.* (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020). rc VP'd \R4h\VPVPVP VP VPVP4#r) rUrrr"r$r&r#r%r'rs&,r)rCG.doitsG    => >dggtww$''477SSr+c4VPVPVPVPVP3RR7pVPVP VP 3RR7p\VP4VP44p\VPR4!p\VPR4!pWSP48Xg4\VPRWSP4, ,4!pWTP48Xg4\VPRWTP4, ,4!p\RRV,,4p\VPV4!p\VPV4!pV#),) delimiterrcC) _print_seqr"r#r$r%r&r'rgr[rrirhrrjabove)r1rmr(bottoppadrss&&* r)rx CG._prettys)  WWdggtww 0C!A  $''477!3s C#))+syy{+#((3-(#((3-(iik!ciiS99;->(?@ACiik!ciiS99;->(?@AC sSW} %  %  %r+c \VPVPVPVPVP VP VP34pR\V4,#)zC^{%s,%s}_{%s,%s,%s,%s}) rrer&r'r"r#r$r%r{r|s&&* r)r~ CG._latexsKGNNTWWdggtwwGGTWWdgg%/0)E%L88r+rN) rrrrrr precedencerrxr~rrrs@r)rrs15lE"Q&JT $99r+ca]tRt^toRtRt]R4t]R4t]R4t ]R4t ]R4t ]R4t ]R 4t R tR tR tR tVtR#)rzQClass for the Wigner-6j symbols See Also ======== Wigner3j: Wigner-3j symbols c X\\WW4WV34p\P!V.VO5!#rr)r!r"r$j12r&r^j23r(s&&&&&&& r)r Wigner6j.__new__s)7RSa56||C'$''r+c(VP^,#r-r/r0s&r)r" Wigner6j.j1r3r+c(VP^,#r5r/r0s&r)r$ Wigner6j.j2r3r+c(VP^,#r9r/r0s&r)r Wigner6j.j12 r3r+c(VP^,#r=r/r0s&r)r& Wigner6j.j3r3r+c(VP^,#rAr/r0s&r)r^ Wigner6j.jr3r+c(VP^,#rEr/r0s&r)r Wigner6j.j23r3r+c\;QJd0RVP4F'dK R'*# R'*#!RVP44'*#)c38"TFqPxK R#5irrJrLs& r)rO'Wigner6j.is_symbolic..rQrRFTrSr0s&r)rUWigner6j.is_symbolicrWr+cdaaVPVP4VPVP43VPVP4VPVP43VPVP 4VPVP 433o^p^pR.^,p\^4F%o\VV3Rl\^444VS&K' Rp\^4EFpRp\^4FoSS,V,p VS,V P4, p V ^,p W, p \V PRV ,4!p \V PRV ,4!p VfT pK\VPRV,4!p\VPV 4!pK VfTpK\V4Fp \VPR4!pK \VPV4!pEK! \VPRRR7!pV#)r:c3`<"TF#pSS,V,P4xK% R#5irrZr\s& r)rO#Wigner6j._pretty..)rarbNrc{}rirhrd)rer"r&r$r^rrrfrgr[rrhrirjrkrls&&* @@r)rxWigner6j._pretty!snnTWW%w~~dgg'> ? ^^DGG $gnnTVV&< = ^^DHH %w~~dhh'? @ BtAvqA<58< >$''488TWWdffdhhOOr+rN)rrrrrr rr"r$rr&r^rrUrxr~rrrrs@r)rrs(;;!F PPr+ca]tRtRtoRtRt]R4t]R4t]R4t ]R4t ]R4t ]R 4t ]R 4t ]R 4t]R 4t]R 4tRtRtRtRtVtR#)riPzQClass for the Wigner-9j symbols See Also ======== Wigner3j: Wigner-3j symbols c \\\WW4WVWxV 3 4p \P!V.V O5!#rr) r!r"r$rr&j4j34j13j24r^r(s &&&&&&&&&& r)r Wigner9j.__new__Ys-7RSbsCD||C'$''r+c(VP^,#r-r/r0s&r)r" Wigner9j.j1]r3r+c(VP^,#r5r/r0s&r)r$ Wigner9j.j2ar3r+c(VP^,#r9r/r0s&r)r Wigner9j.j12er3r+c(VP^,#r=r/r0s&r)r& Wigner9j.j3ir3r+c(VP^,#rAr/r0s&r)r Wigner9j.j4mr3r+c(VP^,#rEr/r0s&r)r Wigner9j.j34qr3r+c(VP^,#)r/r0s&r)r Wigner9j.j13ur3r+c(VP^,#)r/r0s&r)r Wigner9j.j24yr3r+c(VP^,#)r/r0s&r)r^ Wigner9j.j}r3r+c\;QJd0RVP4F'dK R'*# R'*#!RVP44'*#)c38"TFqPxK R#5irrJrLs& r)rO'Wigner9j.is_symbolic..rQrRFTrSr0s&r)rUWigner9j.is_symbolicrWr+caaVPVP4VPVP4VPVP43VPVP4VPVP 4VPVP 43VPVP4VPVP4VPVP433o^p^pR.^,p\^4F%o\VV3Rl\^444VS&K' Rp\^4EFpRp\^4FoSS,V,p VS,V P4, p V ^,p W, p \V PRV ,4!p \V PRV ,4!p VfT pK\VPRV,4!p\VPV 4!pK VfTpK\V4Fp \VP!R4!pK \VP!V4!pEK! \VP#RRR7!pV#)r:c3`<"TF#pSS,V,P4xK% R#5irrZr\s& r)rO#Wigner9j._pretty..rarbNrcrrrrd)rer"r&rr$rrrrr^rfrgr[rrhrirjrkrls&&* @@r)rxWigner9j._prettys ^^!..17>>$((3K M ^^!..17>>$((3K M ^^DHH %w~~dhh'?PTPVPVAW X  Z tAvqA<58< >$''488TWWdggtxxQUQYQY[_[c[ceiekekllr+rN)rrrrrr rr"r$rr&rrrrr^rUrxr~rrrrs@r)rrPs(;;$L mmr+c\V\4'd \V4#\V\4'd \ V4#\V\ 4'd-\ VP Uu.uFp\V4NK up!#\V\4'd*\\VP4VP4#V#uupi)aoSimplify and combine CG coefficients. Explanation =========== 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. ) isinstancer _cg_simp_addr _cg_simp_sumrr(cg_simprbaseexp)erNs& r)rrsB!SA As  A As  QVV4VcWS\V455 As  7166?AEE** 5s,CcL.p.p\V4pVPEFpVP\4'd\ V\ 4'dVP \V44KQ\ V\4'd^pVPF5p\ V\ 4'dV\V4,pK-WE,pK7 VP\4'dVP V4KVP V4KVP V4EKVP V4EK \V4wrVP V4\V4wrVP V4\V4wrVP V4\V!\V!,#)aTakes a sum of terms involving Clebsch-Gordan coefficients and simplifies the terms. Explanation =========== First, we create two lists, cg_part, which is all the terms involving CG coefficients, and other_part, which is all other terms. The cg_part list is then passed to the simplification methods, which return the new cg_part and any additional terms that are added to other_part ) rr(hasrrrappendrr_check_varsh_871_1_check_varsh_871_2_check_varsh_872_9r)rcg_part other_partrNtermstermothers& r)rrs<GJq Avv 772;;#s##!!,s"34C%%HHD!$,,d!33  % 99R==NN5)%%e,s#   c "%((0NGe'0NGe'0NGe =3 + ++r+c \\R4wrr4V\WV^W4,p^V,^,\V^4,pV\ V4, p^V,^,pW,p \ WVWtWW#V3W3W4 #)a)ralphablt)rr rrabs_check_cg_simp) term_listrrrrexprsimpsign build_expr index_exprs & r)rrs$ 9:OAa b1a* *D aC!G^Aq) )D c"g:D1qJJ $d u;LqfV` mmr+c J\\R4wrr4V\WW)V^4,p\^V,^,4\ V^4,pRW, ,V,\ V4, p^V,^,pW,p \ WVWtWW#V3W3W4 #)r)rrcrrd)rr rr rrr) rrrr rrrrr r s & r)rrs$ 9:OAa b1fa+ +D !a=1- -D !) R B 'D1qJJ $d u;LqfV` mmr+c\\R4w rr4rVrxp V \WWEWx4^,,p \Pp V \ V 4, p \ W, 4p \ W%,4pW,^,\ WV83^\W43WV 834, pW,V, p\WWWW$WWW3WWE3VV4 wpp\ W, 4p W,pV^,V , W,^,,pW}, W,,V,V,p\WWWW$WWW3WWE3VV4 wpp\WWEWx4\WWFWx4,p \W#4\WV4,p \Pp \ W, 4p \ W%,4pW,^,\ WV83^\W43WV 834, pW,V, p\WV \PWW#WEWgV3WW4WV3VV4 wpp\ W, 4p W,pV^,V , W,^,,pW}, W,,V,V,p\WV \PWW#WEWgV3WW4WV3VV4 wppVVV,V,3#)r) rralphaprbetabetapr gammar) rr rr Onerrrrr)rrrrrrrr rrrrrxyr r other1other2other3other4s& r)rr)s58@J6K2Afa b1A-q0 0D 55D c"g:D AE A ELAYq5zAr!x=1!e*MMJJ&t4YEVZ_dHilmvwk~AKMWXIv AE A Aa%!)aeai(J%!%1$u,J&t4YEVZ_dHilmvwk~AKMWXIv a *2a1+L LD % ()D DD 55D AE A ELAYq5zAr!x=1!e*MMJJ&t4 u^_glqvKwz{EKPTz\^hjtuIv AE A Aa%!)aeai(J%!%1$u,J&t4 u^_glqvKwz{EKPTz\^hjtuIv fvo. ..r+c h^p ^p V \V48Ed\WJ,V\V44p V f V ^, p K<VPV 4P'g V ^, p KhVU u.uF qW,3NK p p R.VPV 4,p\ V \V44EFp\WO,VPV 4\V4\V4, VPV 4VPV 43R7pVfKgVPV 4PV4P'gKWPV^4PV 4PV4VPV4VPV 4PV43WPV 4PV4&EK \ ;QJdRV4F 'gK RM RM !RV44'Eg\ VUu.uFp\V^,4NK up!pVUu.uF pV^,NK ppVP4VP4VUu.uFqPV4NK pVFWp\V^,4V8gKVPV^,VV^,,, V^,,4KY V VW!,PV 4,, p EK V ^, p EKWI3#uup iuupiuupiuupi)aLChecks for simplifications that can be made, returning a tuple of the simplified list of terms and any terms generated by simplification. Parameters ========== expr: expression The expression with Wild terms that will be matched to the terms in the sum simp: expression The expression with Wild terms that is substituted in place of the CG terms in the case of simplification sign: expression The expression with Wild terms denoting the sign that is on expr that must match lt: expression The expression with Wild terms that gives the leading term of the matched expr term_list: list A list of all of the terms is the sum to be simplified variables: list A list of all the variables that appears in expr dep_variables: list A list of the variables that must match for all the terms in the sum, i.e. the dependent variables build_index_expr: expression Expression with Wild terms giving the number of elements in cg_index index_expr: expression Expression with Wild terms giving the index terms have when storing them to cg_index N)rc3("TFqRJxK R#5irr)rMr]s& r)rO!_check_cg_simp..s/h9hsTF) len _check_cgsubsrKrfanyminrsortreversepopr)rrrrr variables dep_variablesbuild_index_exprr rr]sub_1rsub_depcg_indexr^sub_2rmin_ltindicess&&&&&&&&& r)rrVsRJ A c)n ),c)n= = FA $$U+555 FA *78-Qux=-86*//66q#i.)AilDIIg,>IQTUbQc@ckoktktuzk{~B~G~GHO~PkQRE}??7+007AAA=> "a@P@U@UV]@^@c@cdi@jlnlslstylz}A}F}FGN}O}T}TUZ}[>[H__W-2259 : *s/h/sss/h///X?XTCQLX?@F,45HDQHG5 LLN OO (/ 11mmA 1 tAw<&($$tAwQ'?a&HJ! &$)!1!1%!88 8J FA   +9@5 2s1L L%9L*1L/NcVPV4pVfR#VeH\V\4'g \R4hV^,V^,P V48XgR#\ V4V8XdV#R#)z2Checks whether a term matches the given expressionNzsign must be a tuple)matchrr{ TypeErrorrr)cg_termrlengthrmatchess&&&& r)rrskmmD!G $&&23 3Aw47..11  7|vr+cH\V4p\V4p\V4pV#r)_check_varsh_sum_871_1_check_varsh_sum_871_2_check_varsh_sum_872_4)rs&r)rrs%q!Aq!Aq!A Hr+c $\R4p\R4p\R4pVP\\ WV^W4W!)V344pVeA\ V4^8Xd1^V,^,\ V^4,PV4#V#)rrr)r r r.rrrrr)rrrrr.s& r)r4r4s~ S A G E S A GGC1Q14ub!nE FE SZ1_1q.A..44U;; Hr+c `\R4p\R4p\R4pVP\RW, ,\ WW)V^4,W!)V344pVeJ\ V4^8Xd:\ ^V,^,4\V^4,PV4#V#)rrr rd) r r r.rrrr rr)rrrr r.s& r)r5r5s S A G E S A GG R19 b1fa; ;eR^L NE SZ1_QqS1W nQ2288?? Hr+c P\R4p\R4p\R4p\R4p\R4p\R4p\R4p\R4p\W1WBWW4p \W1WBWh4p VP\ W,W)V3W$)V344p V e;\ V 4^8Xd+\ WV4\ Wx4,PV 4#VP\ V ^,W)V3W$)V344p V e!\ V 4^8Xd\P#V#)rrrrr cprgammap) r r rr.rrrrr r) rrrrrr r:rr;cg1cg2match1match2s & r)r6r6s G E 6?D S A S A S A dB ME (^F Qq )C Qq +C WWS5"a.4Q-@ AF c&kQ.q%nU&CCII&QQ WWSa%Q$A? @F c&kQ.uu Hr+c\V\4'dV3^^3#.p^p\V\\34'g \ R4h\V\4'd|VP P 'd`VP P 'd>\VP 4Uu.uFq1PVP4NK pMV3^^3#\V\4'dYVPF4p\V\4'dVPV4K,W$,pK6 WV\V4, 3#R#uupi)r6z term must be CG, Add, Mul or PowN) rrrrNotImplementedErrorrrKrfrrr(r)rcgcoeffrwrNs& r)_cg_listrDs$w1} B E dS#J ' '!"DEE$!3!3!3 88   ,1$((O =Oqii "O =7Aq= $99C#r"" #   %E ***  >s-#E )rrrrrr)7rsympy.concrete.summationsrsympy.core.addrsympy.core.exprrsympy.core.functionrsympy.core.mulrsympy.core.powerrsympy.core.relationalrsympy.core.singletonr sympy.core.symbolr r sympy.core.sympifyr (sympy.functions.elementary.miscellaneousr $sympy.functions.elementary.piecewiser sympy.printing.pretty.stringpictrr(sympy.functions.special.tensor_functionsrsympy.physics.wignerrrrrsympy.printing.precedencer__all__rrrrrrrrrrrrr4r5r6rDrr+r)rVs #) & $"-&9:CCPP0 xOtxOvS9S9lVPtVPremtemP*Z+,\nn*/ZH!V     (+r+