\h[SrSSKJr SSKJr SSKJr SSKJr SSK J r SSK J r SSK Jr SS KJr SS KJrJr SS KJr SS KJr SS KJr SSKJrJr SSKJr SSKJ r J!r!J"r"J#r# SSK$J%r% /SQr&"SS\5r'"SS\'5r("SS\5r)"SS\5r*Sr+Sr,Sr-Sr.Sr/S r0S(S"jr1S#r2S$r3S%r4S&r5S'r6g!))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) PRECEDENCE)CGWigner3jWigner6jWigner9jcg_simpc\rSrSrSrSrSr\S5r\S5r \S5r \S5r \S 5r \S 5r \S 5rS rS rSrSrg)r&a8Class 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 V[[XX4XV45n[R"U/UQ76$Nmapr r__new__)clsj1m1j2m2j3m3argss P/var/www/auris/envauris/lib/python3.13/site-packages/sympy/physics/quantum/cg.pyr#Wigner3j.__new__Qs)7RRR45||C'$''c URS$Nrr+selfs r,r% Wigner3j.j1Uyy|r.c URS$Nr1r2s r,r& Wigner3j.m1Yr5r.c URS$Nr1r2s r,r' Wigner3j.j2]r5r.c URS$Nr1r2s r,r( Wigner3j.m2ar5r.c URS$Nr1r2s r,r) Wigner3j.j3er5r.c URS$Nr1r2s r,r* Wigner3j.m3ir5r.cD[SUR55(+$)Nc38# UHoRv M g7fr  is_number.0args r, 'Wigner3j.is_symbolic..o: }} allr+r2s r, is_symbolicWigner3j.is_symbolicm: ::::r.c^^URUR5URUR54URUR5URUR54URUR 5URUR 544mSnSnS/S-n[S5H%m[UU4Sj[S555UT'M' Sn[S5HnSn[S5HmTTUn UTU R5- n U S-n X- n [U RSU -56n [U RSU -56n UcU nMe[URSU-56n[URU 56nM UcUnM[U5Hn [URS56nM [URU56nM [UR56nU$)Nr<r8r@c3P># UHnTTUR5v M g7fr widthrOijms r,rQ#Wigner3j._pretty..z!<8a!A$q'--//8#& )_printr%r&r'r(r)r*rangemaxr^rrightleftbelowparensr3printerr+hsepvsepmaxwDr`D_rowswdeltawleftwright_rarbs @@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*rWrzrr__static_attributes__rr.r,rr&s&PN(;;!F Or.rc:\rSrSrSr\SS- rSrSrSr Sr g ) 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). rr8c UR(a [S5e[URURUR UR URUR5$r) rWrrr%r'r)r&r(r*rs r,rCG.doitsD   => >dggtww$''477SSr.cURURURURUR4SS9nURUR UR 4SS9n[UR5UR55n[URS56n[URS56nXSR5:Xd*[URSXSR5- -56nXTR5:Xd*[URSXTR5- -56n[SSU--5n[URU56n[URU56nU$)N,) delimiterrfC) _print_seqr%r&r'r(r)r*rir^rrkrjrrlabove)r3ror+bottoppadrus r,rz CG._prettys)  WWdggtww 0C!A  $''477!3s C#))+syy{+#((3-(#((3-(iik!ciiS99;->(?@ACiik!ciiS99;->(?@AC sSW} %  %  %r.c [URURURURUR UR UR45nS[U5-$)NzC^{%s,%s}_{%s,%s,%s,%s}) r"rgr)r*r%r&r'r(r}r~s r,r CG._latexsKGNNTWWdggtwwGGTWWdgg%/0)E%L88r.rN) rrrrrr precedencerrzrrrr.r,rrs)5lE"Q&JT $9r.rc\rSrSrSrSr\S5r\S5r\S5r \S5r \S5r \S 5r \S 5r S rS rS rSrg)rzQClass for the Wigner-6j symbols See Also ======== Wigner3j: Wigner-3j symbols c V[[XX4XV45n[R"U/UQ76$r r!)r$r%r'j12r)raj23r+s r,r#Wigner6j.__new__s)7RSa56||C'$''r.c URS$r0r1r2s r,r% Wigner6j.j1r5r.c URS$r7r1r2s r,r' Wigner6j.j2r5r.c URS$r;r1r2s r,r Wigner6j.j12 r5r.c URS$r?r1r2s r,r) Wigner6j.j3r5r.c URS$rCr1r2s r,ra Wigner6j.jr5r.c URS$rGr1r2s r,r Wigner6j.j23r5r.cD[SUR55(+$)Nc38# UHoRv M g7fr rLrNs r,rQ'Wigner6j.is_symbolic..rSrTrUr2s r,rWWigner6j.is_symbolicrYr.c^^URUR5URUR54URUR5URUR54URUR 5URUR 544mSnSnS/S-n[S5H%m[UU4Sj[S555UT'M' Sn[S5HnSn[S5HmTTUn UTU R5- n U S-n X- n [U RSU -56n [U RSU -56n UcU nMe[URSU-56n[URU 56nM UcUnM[U5Hn [URS56nM [URU56nM [URSSS 96nU$) Nr<r8r[r@c3P># UHnTTUR5v M g7fr r]r_s r,rQ#Wigner6j._pretty..)rdrerf{}rkrj)rgr%r)r'rarrrhrir^rrjrkrlrmrns @@r,rzWigner6j._pretty!snnTWW%w~~dgg'> ? ^^DGG $gnnTVV&< = ^^DHH %w~~dhh'? @ BtAvqA<58< >$''488TWWdffdhhOOr.rN)rrrrrr#rr%r'rr)rarrWrzrrrrr.r,rrs(;;!F Pr.rc\rSrSrSrSr\S5r\S5r\S5r \S5r \S5r \S 5r \S 5r \S 5r\S 5r\S 5rSrSrSrSrg)riPzQClass for the Wigner-9j symbols See Also ======== Wigner3j: Wigner-3j symbols c Z[[XX4XVXxU 4 5n [R"U/U Q76$r r!) r$r%r'rr)j4j34j13j24rar+s r,r#Wigner9j.__new__Ys-7RSbsCD||C'$''r.c URS$r0r1r2s r,r% Wigner9j.j1]r5r.c URS$r7r1r2s r,r' Wigner9j.j2ar5r.c URS$r;r1r2s r,r Wigner9j.j12er5r.c URS$r?r1r2s r,r) Wigner9j.j3ir5r.c URS$rCr1r2s r,r Wigner9j.j4mr5r.c URS$rGr1r2s r,r Wigner9j.j34qr5r.c URS$)Nr1r2s r,r Wigner9j.j13ur5r.c URS$)Nr1r2s r,r Wigner9j.j24yr5r.c URS$)Nr1r2s r,ra Wigner9j.j}r5r.cD[SUR55(+$)Nc38# UHoRv M g7fr rLrNs r,rQ'Wigner9j.is_symbolic..rSrTrUr2s r,rWWigner9j.is_symbolicrYr.c^^URUR5URUR5URUR54URUR5URUR 5URUR 54URUR5URUR5URUR544mSnSnS/S-n[S5H%m[UU4Sj[S555UT'M' Sn[S5HnSn[S5HmTTUn UTU R5- n U S-n X- n [U RSU -56n [U RSU -56n UcU nMe[URSU-56n[URU 56nM UcUnM[U5Hn [UR!S56nM [UR!U56nM [UR#SSS 96nU$) Nr<r8r[r@c3P># UHnTTUR5v M g7fr r]r_s r,rQ#Wigner9j._pretty..rdrerfrrr)rgr%r)rr'rrrrrarhrir^rrjrkrlrmrns @@r,rzWigner9j._prettys ^^!..17>>$((3K M ^^!..17>>$((3K M ^^DHH %w~~dhh'?PTPVPVAW X  Z tAvqA<58< >$''488TWWdggtxxQUQYQY[_[c[ceiekekllr.rN)rrrrrr#rr%r'rr)rrrrrarWrzrrrrr.r,rrPs(;;$L mr.rc[U[5(a [U5$[U[5(a [ U5$[U[ 5(a,[ UR Vs/sHn[U5PM sn6$[U[5(a)[[UR5UR5$U$s snf)aoSimplify and combine CG coefficients. Explanation =========== This function uses various symmetry and properties of sums and products of Clebsch-Gordan coefficients to simplify statements involving these terms [1]_. Examples ======== Simplify the sum over CG(a,alpha,0,0,a,alpha) for all alpha to 2*a+1 >>> from sympy.physics.quantum.cg import CG, cg_simp >>> a = CG(1,1,0,0,1,1) >>> b = CG(1,0,0,0,1,0) >>> c = CG(1,-1,0,0,1,-1) >>> cg_simp(a+b+c) 3 See Also ======== CG: Clebsh-Gordan coefficients References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. ) isinstancer _cg_simp_addr _cg_simp_sumrr+rrbaseexp)erPs r,rrsB!SA As  A As  QVV4VcWS\V455 As  7166?AEE** 5s)Cc$/n/n[U5nURGHnUR[5(a[ U[ 5(aUR [U55 MO[ U[5(a~SnURH,n[ U[ 5(aU[U5-nM(XE-nM. UR[5(aUR U5 MUR U5 MUR U5 MUR U5 GM [U5upUR U5 [U5upUR U5 [U5upUR U5 [U6[U6-$)aTakes a sum of terms involving Clebsch-Gordan coefficients and simplifies the terms. Explanation =========== First, we create two lists, cg_part, which is all the terms involving CG coefficients, and other_part, which is all other terms. The cg_part list is then passed to the simplification methods, which return the new cg_part and any additional terms that are added to other_part r8) rr+hasrrrappendrr_check_varsh_871_1_check_varsh_871_2_check_varsh_872_9r)rcg_part other_partrPtermstermothers r,rrs<GJq Avv 772;;#s##!!,s"34C%%HHD!$,,d!33  % 99R==NN5)%%e,s#   c "%((0NGe'0NGe'0NGe =3 + ++r.c [[S5upp4U[XUSX5-nSU-S-[US5-nU[ U5- nSU-S-nX-n [ XVXtXX#U4X4X5 $)N)aalphabltrr<r8)r"r rrabs_check_cg_simp) term_listrrrrexprsimpsign build_expr index_exprs r,rrs$ 9:OAa b1a* *D aC!G^Aq) )D c"g:D1qJJ $d u;LqfV` mmr.c [[S5upp4U[XX*US5-n[SU-S-5[ US5-nSX- -U-[ U5- nSU-S-nX-n [ XVXtXX#U4X4X5 $)N)rrcrrr<r8r[)r"r rrrrr) rrrr rrrrr r s r,rrs$ 9:OAa b1fa+ +D !a=1- -D !) R B 'D1qJJ $d u;LqfV` mmr.c[[S5u pp4pVpxn U [XXEXx5S--n [Rn U [ U 5- n [ X- 5n [ X%-5nX-S-[ XU:4S[X54XU :45- nX-U- n[XXXX$XWX4XXE4UU5 unn[ X- 5n X-nUS-U - X-S--nX}- X--U-U-n[XXXX$XWX4XXE4UU5 unn[XXEXx5[XXFXx5-n [X#5[XV5-n [Rn [ X- 5n [ X%-5nX-S-[ XU:4S[X54XU :45- nX-U- n[XU [RXX#XEXgU4XX4XV4UU5 unn[ X- 5n X-nUS-U - X-S--nX}- X--U-U-n[XU [RXX#XEXgU4XX4XV4UU5 unnUUU-U-4$)N) rralphaprbetabetapr gammarr<r8r) r"r rr Onerrr rr)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 Sn Sn U [U5:Ga[XJU[U55n U cU S- n M2URU 5R(dU S- n MYUV s/sHoX4PM n n S/URU 5-n[ U [U55GH n[XOURU 5[U5[U5- URU 5URU 54S9nUcM[URU 5RU5R(dMXRUS5RU 5RU5URU5URU 5RU54XRU 5RU5'GM [ SU55(d[ UVs/sHn[US5PM sn6nUVs/sHnUSPM nnUR5 UR5 UVs/sHoRU5PM nUH:n[US5U:dMURUSUUS-- US-5 M< U UX!-RU 5-- n OU S- n U [U5:aGMXI4$s sn fs snfs snfs snf)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 rNr8)rc3(# UHoSLv M g7fr r)rOr`s r,rQ!_check_cg_simp..s/h9hsr<r@) len _check_cgsubsrMrhanyminrsortreversepopr)rrrrr variables dep_variablesbuild_index_exprr rr`sub_1rsub_depcg_indexrasub_2rmin_ltindicess r,rrVsRJ A c)n ),c)n= = FA $$U+55 FA *78-Qux=-86*//66q#i.)AilDIIg,>IQTUbQc@ckoktktuzk{~B~G~GHO~PkQRE}??7+007AA=> "a@P@U@UV]@^@c@cdi@jlnlslstylz}A}F}FGN}O}T}TUZ}[>[H__W-2259 : */h///X?XTCQLX?@F,45HDQHG5 LLN OO (/ 11mmA 1 tAw<&($$tAwQ'?a&HJ! &$)!1!1%!88 8J FA9 c)n :   +9@5 2s"J3J8.J=#KNcURU5nUcgUb<[U[5(d [S5eUSUSR U5:Xdg[ U5U:XaU$g)z2Checks whether a term matches the given expressionNzsign must be a tuplerr8)matchrr} TypeErrorrr)cg_termrlengthrmatchess r,rrskmmD!G $&&23 3Aw47..11  7|vr.cH[U5n[U5n[U5nU$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 [S5n[S5n[S5nUR[[ XUSX5X!*U455nUb3[ U5S:Xa$SU-S-[ US5-RU5$U$)Nrrrrr<r8)r r r.rrrrr)rrrrr.s r,r4r4s~ S A G E S A GGC1Q14ub!nE FE SZ1_1q.A..44U;; Hr.c ,[S5n[S5n[S5nUR[SX- -[ XX*US5-X!*U455nUb<[ U5S:Xa-[ SU-S-5[US5-RU5$U$)Nrrr r[rr<r8) r r r.rrrrrr)rrrr r.s r,r5r5s S A G E S A GG R19 b1fa; ;eR^L NE SZ1_QqS1W nQ2288?? Hr.c 0[S5n[S5n[S5n[S5n[S5n[S5n[S5n[S5n[X1XBXW5n [X1XBXh5n UR[ X-X*U4X$*U455n U b5[ U 5S :Xa&[ XV5[ Xx5-RU 5$UR[ U S -X*U4X$*U455n U b[ U 5S :Xa[R$U$) Nrrrrr cprgammaprr<rD) 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$A? @F c&kQ.uu Hr.c~[U[5(aU4SS4$/nSn[U[[45(d [ S5e[U[5(azUR R (a_UR R (a>[UR 5Vs/sHo1RUR5PM nOU4SS4$[U[5(aOURH/n[U[5(aURU5 M+X$-nM1 XU[U5- 4$gs snf)Nr8z term must be CG, Add, Mul or Pow) rrrrNotImplementedErrorrrMrhrrr+r)rcgcoeffryrPs r,_cg_listrDs$w1} B E dS#J ' '!"DEE$!3!3 88  ,1$((O =Oqii "O =7Aq= $99C#r"" #   %E ***  >s($D:r )7rsympy.concrete.summationsrsympy.core.addrsympy.core.exprrsympy.core.functionrsympy.core.mulrsympy.core.powerrsympy.core.relationalr sympy.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.