Ed"[dZddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZmZdd lmZdd lmZdd lmZddlmZmZddlmZddlm Z m!Z!m"Z"m#Z#ddl$m%Z%gdZ&GddeZ'Gdde'Z(GddeZ)GddeZ*dZ+dZ,dZ-dZ.dZ/d Z0d(d"Z1d#Z2d$Z3d%Z4d&Z5d'Z6d!S))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_simpceZdZdZdZdZedZedZedZ edZ edZ ed Z ed Z d Zd Zd ZdS)raClass 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 \tt||||||f}tj|g|RSNmapr r__new__)clsj1m1j2m2j3m3argss 8lib/python3.11/site-packages/sympy/physics/quantum/cg.pyr"zWigner3j.__new__Qs67RRR455|C'$''''c|jdSNrr*selfs r+r$z Wigner3j.j1Uy|r,c|jdSNr/r0s r+r%z Wigner3j.m1Yr2r,c|jdSNr/r0s r+r&z Wigner3j.j2]r2r,c|jdSNr/r0s r+r'z Wigner3j.m2ar2r,c|jdSNr/r0s r+r(z Wigner3j.j3er2r,c|jdSNr/r0s r+r)z Wigner3j.m3ir2r,c@td|jD S)Nc3$K|] }|jV dSr is_number.0args r+ z'Wigner3j.is_symbolic..o$::s}::::::r,allr*r0s r+ is_symboliczWigner3j.is_symbolicm$:: :::::::r,c^||j||jf||j||jf||j||jffd}d}dgdz}tdD].tfdtdD|</d}tdD]}d}tdD]|} || z } | dz} | | z } t| d| z} t| d| z} || }zt| d|z}t| | }||}t|D]} t| d}t| |}t|}|S)Nr8r5r;cPg|]"}|#SwidthrGijms r+ z$Wigner3j._pretty..z)???AaDGMMOO???r, )_printr$r%r&r'r(r)rangemaxrTrrightleftbelowparensr1printerr*hsepvsepmaxwDrVD_rowswdeltawleftwright_rWrXs @@r+_prettyzWigner3j._prettyrsnnTW%%w~~dg'>'> ? ^^DG $ $gnnTW&=&= > ^^DG $ $gnnTW&=&= > @tAvq A AA?????U1XX???@@DGG q , ,AE1XX 4 4aDGa17799, %F 3 34s5y 1 12E"EKKD$9$9:"EKKNN3 4[[ . . -AGGENN+AA  #r,c t|j|j|j|j|j|j|jf}dt|zS)NzH\left(\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right)) r!r\r$r&r(r%r'r)tupler1rdr*labels r+_latexzWigner3j._latexsIGNTWdgtwGTWdg%/00Z %LL r,c |jrtdt|j|j|j|j|j|jSNzCoefficients must be numerical) rM ValueErrorrr$r&r(r%r'r)r1hintss r+doitz Wigner3j.doits@   ?=>> >$'47DGTWdgNNNr,N)__name__ __module__ __qualname____doc__is_commutativer"propertyr$r%r&r'r(r)rMrortrzrRr,r+rr&s%&&PN(((XXXXXX;;X;!!!F OOOOOr,rc:eZdZdZeddz ZdZdZdZdS)raClass for Clebsch-Gordan coefficient. Explanation =========== Clebsch-Gordan coefficients describe the angular momentum coupling between two systems. The coefficients give the expansion of a coupled total angular momentum state and an uncoupled tensor product state. The Clebsch-Gordan coefficients are defined as [1]_: .. math :: C^{j_3,m_3}_{j_1,m_1,j_2,m_2} = \left\langle j_1,m_1;j_2,m_2 | j_3,m_3\right\rangle Parameters ========== j1, m1, j2, m2 : Number, Symbol Angular momenta of states 1 and 2. j3, m3: Number, Symbol Total angular momentum of the coupled system. Examples ======== Define a Clebsch-Gordan coefficient and evaluate its value >>> from sympy.physics.quantum.cg import CG >>> from sympy import S >>> cg = CG(S(3)/2, S(3)/2, S(1)/2, -S(1)/2, 1, 1) >>> cg CG(3/2, 3/2, 1/2, -1/2, 1, 1) >>> cg.doit() sqrt(3)/2 >>> CG(j1=S(1)/2, m1=-S(1)/2, j2=S(1)/2, m2=+S(1)/2, j3=1, m3=0).doit() sqrt(2)/2 Compare [2]_. See Also ======== Wigner3j: Wigner-3j symbols References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. .. [2] `Clebsch-Gordan Coefficients, Spherical Harmonics, and d Functions `_ in P.A. Zyla *et al.* (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020). rr5c |jrtdt|j|j|j|j|j|jSrv) rMrwrr$r&r(r%r'r)rxs r+rzzCG.doits@   ?=>> >dgtw$'47SSSr,cL||j|j|j|jfd}||j|jfd}t||}t| d}t| d}||ks4t| d||z z}||ks4t| d||z z}tdd|zz}t| |}t||}|S)N,) delimiterr[C) _print_seqr$r%r&r'r(r)r^rTrr`r_rraabove)r1rdr*bottoppadrjs r+roz CG._prettysZ  Wdgtw 0C!AA  $'47!3s CC#))++syy{{++#((3--(#((3--(ciikk! BciiS399;;->(?@@ACciikk! BciiS399;;->(?@@AC sSW} % %  %  %r,c t|j|j|j|j|j|j|jf}dt|zS)NzC^{%s,%s}_{%s,%s,%s,%s}) r!r\r(r)r$r%r&r'rqrrs r+rtz CG._latexsDGNTWdgtwGTWdg%/00)E%LL88r,N) r{r|r}r~r precedencerzrortrRr,r+rrs`55lE"Q&JTTT $99999r,rceZdZdZdZedZedZedZedZ edZ edZ ed Z d Z d Zd Zd S)rzaClass for the Wigner-6j symbols See Also ======== Wigner3j: Wigner-3j symbols c \tt||||||f}tj|g|RSrr )r#r$r&j12r(rWj23r*s r+r"zWigner6j.__new__s67RS"a566|C'$''''r,c|jdSr.r/r0s r+r$z Wigner6j.j1r2r,c|jdSr4r/r0s r+r&z Wigner6j.j2r2r,c|jdSr7r/r0s r+rz Wigner6j.j12 r2r,c|jdSr:r/r0s r+r(z Wigner6j.j3r2r,c|jdSr=r/r0s r+rWz Wigner6j.jr2r,c|jdSr@r/r0s r+rz Wigner6j.j23r2r,c@td|jD S)Nc3$K|] }|jV dSrrDrFs r+rIz'Wigner6j.is_symbolic..rJr,rKr0s r+rMzWigner6j.is_symbolicrNr,cd||j||jf||j||jf||j||jffd}d}dgdz}tdD].tfdtdD|</d}tdD]}d}tdD]|} || z } | dz} | | z } t| d| z} t| d| z} || }zt| d|z}t| | }||}t|D]} t| d}t| |}t|dd }|S) Nr8r5rPr;cPg|]"}|#SrRrSrUs r+rYz$Wigner6j._pretty..)rZr,r[{}r`r_)r\r$r(r&rWrrr]r^rTrr_r`rarbrcs @@r+rozWigner6j._pretty!s$nnTW%%w~~dg'>'> ? ^^DG $ $gnnTV&<&< = ^^DH % %w~~dh'?'? @ BtAvq A AA?????U1XX???@@DGG q , ,AE1XX 4 4aDGa17799, %F 3 34s5y 1 12E"EKKD$9$9:"EKKNN3 4[[ . . -AGGENN+AA c55 6r,c t|j|j|j|j|j|j|jf}dt|zS)NzJ\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right\}) r!r\r$r&rr(rWrrqrrs r+rtzWigner6j._latexDsIGNTWdgtxGTVTX%/00\ %LL r,c |jrtdt|j|j|j|j|j|jSrv) rMrwrr$r&rr(rWrrxs r+rzz Wigner6j.doitJs@   ?=>> >$'48TWdfdhOOOr,N)r{r|r}r~r"rr$r&rr(rWrrMrortrzrRr,r+rrs(((XXXXXX;;X;!!!F PPPPPr,rceZdZdZdZedZedZedZedZ edZ edZ ed Z ed Z ed Zed Zd ZdZdZdS)rzaClass for the Wigner-9j symbols See Also ======== Wigner3j: Wigner-3j symbols c btt||||||||| f } tj|g| RSrr ) r#r$r&rr(j4j34j13j24rWr*s r+r"zWigner9j.__new__Ys<7RS"b#sCCDD|C'$''''r,c|jdSr.r/r0s r+r$z Wigner9j.j1]r2r,c|jdSr4r/r0s r+r&z Wigner9j.j2ar2r,c|jdSr7r/r0s r+rz Wigner9j.j12er2r,c|jdSr:r/r0s r+r(z Wigner9j.j3ir2r,c|jdSr=r/r0s r+rz Wigner9j.j4mr2r,c|jdSr@r/r0s r+rz Wigner9j.j34qr2r,c|jdS)Nr/r0s r+rz Wigner9j.j13ur2r,c|jdS)Nr/r0s r+rz Wigner9j.j24yr2r,c|jdS)Nr/r0s r+rWz Wigner9j.j}r2r,c@td|jD S)Nc3$K|] }|jV dSrrDrFs r+rIz'Wigner9j.is_symbolic..rJr,rKr0s r+rMzWigner9j.is_symbolicrNr,c||j||j||jf||j||j||jf||j||j||j ffd}d}dgdz}tdD].tfdtdD|</d}tdD]}d}tdD]|} || z } | dz} | | z } t| d| z} t| d| z} || }zt|d|z}t|| }||}t|D]} t|d}t||}t|dd }|S) Nr8r5rPr;cPg|]"}|#SrRrSrUs r+rYz$Wigner9j._pretty..rZr,r[rrr)r\r$r(rr&rrrrrWr]r^rTrr_r`rarbrcs @@r+rozWigner9j._prettysh ^^!..117>>$(3K3K M ^^!..117>>$(3K3K M ^^DH % %w~~dh'?'?PTPVAWAW X  Z tAvq A AA?????U1XX???@@DGG q , ,AE1XX 4 4aDGa17799, %F 3 34s5y 1 12E"EKKD$9$9:"EKKNN3 4[[ . . -AGGENN+AA c55 6r,c t|j|j|j|j|j|j|j|j|j |j f }dt|zS)NzZ\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \\ %s & %s & %s \end{array}\right\}) r!r\r$r&rr(rrrrrWrqrrs r+rtzWigner9j._latexsWGNTWdgtx48TXtv%?@@l %LL r,c |jrtdt|j|j|j|j|j|j|j |j |j Srv) rMrwrr$r&rr(rrrrrWrxs r+rzz Wigner9j.doitsR   ?=>> >$'48TWdgtxQUQY[_[ceieklllr,N)r{r|r}r~r"rr$r&rr(rrrrrWrMrortrzrRr,r+rrPsn(((XXXXXXXXX;;X;$$$L mmmmmr,rcht|trt|St|trt |St|t rt d|jDSt|tr'tt|j |j S|S)aSimplify 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. c,g|]}t|SrR)rrFs r+rYzcg_simp..s444cWS\\444r,) isinstancer _cg_simp_addr _cg_simp_sumrr*rrbaseexpes r+rrsB!S A As  A As  44QV44455 As  716??AE***r,crg}g}t|}|jD]}|trt |t r#|t|Ut |trd}|jD]/}t |t r|t|z}*||z}0|tr||||||||t|\}}||t|\}}||t|\}}||t|t|zS)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 r5) rr*hasrrrappendrr_check_varsh_871_1_check_varsh_871_2_check_varsh_872_9r)rcg_part other_partrHtermstermothers r+rrsGJq Av## 772;; ##s## $!!,s"3"34444C%% $H&&D!$,,&d!3!33 99R==-NN5))))%%e,,,,s####   c " " " "'00NGUe'00NGUe'00NGUe =3 + ++r,c ttd\}}}}|t|||d||z}d|zdzt|dz}|t |z }d|zdz}||z} t |||||||||f||f|| S)N)aalphabltrr8r5)r!r rrabs_check_cg_simp) term_listrrrrexprsimpsign build_expr index_exprs r+rrs$ 9::OAua bE1aE** *D aC!G^Aq)) )D c"gg:D1qJUJ $dB Aua;LqRSfV`bl m mmr,c <ttd\}}}}|t|||| |dz}td|zdzt |dz}d||z z|zt |z }d|zdz}||z} t |||||||||f||f|| S)N)rrcrrr8r5rP)r!r rrrrr) rrrrrrrrrrs r+rrs$ 9::OAua bE1ufa++ +D !a==1-- -D !e) R B 'D1qJUJ $dB Aua;LqRSfV`bl m mmr,c8ttd\ }}}}}}}}} | t||||||dzz} tj} | t | z } t ||z } t ||z}||zdzt | | |kfdt| |f||| kfz }||z|z }t| | | | |||||||| f||||f|| \}}t ||z } ||z}|dz| z | |zdzz}|| z | |zz|z|z}t| | | | |||||||| f||||f|| \}}t||||||t||||||z} t||t||z} tj} t ||z } t ||z}||zdzt | | |kfdt| |f||| kfz }||z|z }t| | | tj|||||||||f||||||f|| \}}t ||z } ||z}|dz| z | |zdzz}|| z | |zz|z|z}t| | | tj|||||||||f||||||f|| \}}|||z|zfS)N) rralphaprbetabetaprgammarr8r5r) r!r rr Onerrr rr)rrrrrrrrrrrrrxyrrother1other2other3other4s r+rr)s58@J6K6K2Aufaua bE1dAu--q0 0D 5D c"gg:D AE A EDLAQY1q5zAr!Qxx=1a!e*MMMJQJ&tT4YESTVZ\]_dfhHilmotvwy}k~AKMWXXIv AE A AAa%!)a!eai(Ja%!a%1$u,J&tT4YESTVZ\]_dfhHilmotvwy}k~AKMWXXIv a4E * *2aE1e+L+L LD % ( (e)D)D DD 5D AE A EDLAQY1q5zAr!Qxx=1a!e*MMMJQJ&tT4 AuV\^_aeglnoqvKwz{~CEKMNPTV[z\^hjtuuIv AE A AAa%!)a!eai(Ja%!a%1$u,J&tT4 AuV\^_aeglnoqvKwz{~CEKMNPTV[z\^hjtuuIv fvo. ..r,c d} d} | tkrt| |t|| dz } @|js| dz } `fd|D} dg|z} t | tD]G} t| || t|t|z ||| f}|w|| |js| ||d| ||||| |f| || |<It d| Dst d| D}d| D}||fd |D| D]K}t|d |kr0 |d ||d zz |dzL| |||zzz } n| dz } | tk͉| fS) a Checks for simplifications that can be made, returning a tuple of the simplified list of terms and any terms generated by simplification. Parameters ========== expr: expression The expression with Wild terms that will be matched to the terms in the sum simp: expression The expression with Wild terms that is substituted in place of the CG terms in the case of simplification sign: expression The expression with Wild terms denoting the sign that is on expr that must match lt: expression The expression with Wild terms that gives the leading term of the matched expr term_list: list A list of all of the terms is the sum to be simplified variables: list A list of all the variables that appears in expr dep_variables: list A list of the variables that must match for all the terms in the sum, i.e. the dependent variables build_index_expr: expression Expression with Wild terms giving the number of elements in cg_index index_expr: expression Expression with Wild terms giving the index terms have when storing them to cg_index rNr5c$g|] }||f SrRrR)rGrsub_1s r+rYz"_check_cg_simp..s!888QAuQx=888r,)rc3K|]}|duV dSrrR)rGrVs r+rIz!_check_cg_simp..s&//19//////r,c8g|]}t|dS)r8)rrGrs r+rYz"_check_cg_simp..s"???TCQLL???r,cg|] }|d S)rrRrs r+rYz"_check_cg_simp..s555DQ555r,c:g|]}|SrR)pop)rGrWrs r+rYz"_check_cg_simp..s% 1 1 11immA 1 1 1r,r8r;) len _check_cgsubsrEr]anyminsortreverserr)rrrrr variables dep_variablesbuild_index_exprrrrVsub_depcg_indexrWsub_2min_ltindicesrrs ` @r+rrVsgRJ A c)nn )A,c)nn==   FA $$U++5  FA 8888-8886*//666q#i..)) [ [AilDIIg,>,>IQTUbQcQc@ckoktktuzk{k{~B~G~GHO~P~PkQRRRE ??7++0077A => "a@P@P@U@UV]@^@^@c@cdi@j@jlnlslstylzlz}A}F}FGN}O}O}T}TUZ}[}[>[HZ__W--22599 : ://h///// ??X???@F55H555G LLNNN OO    1 1 1 1 1 1 1 1  K KtAw<<&(K$$tAwQ'?a&HJJJ &$t)!1!1%!8!88 8JJ FA9 c)nn : j  r,Nc||}|dS|Kt|tstd|d|d|ksdSt ||kr|SdS)z2Checks whether a term matches the given expressionNzsign must be a tuplerr5)matchrrq TypeErrorrr)cg_termrlengthrmatchess r+rrsmmD!!G $&& 4233 3Aw47..111  F 7||vr,c`t|}t|}t|}|Sr)_check_varsh_sum_871_1_check_varsh_sum_871_2_check_varsh_sum_872_4rs r+rrs.q!!Aq!!Aq!!A Hr,c Ttd}td}td}|tt |||d|||| |f}|?t |dkr,d|zdzt |dz|S|S)Nrrrrr8r5)r r rrrrrr)rrrrrs r+rrs S A G  E S A GGC1eQ1e44uqb!nEE F FE ) r r rrrrrrr r) rrrrrrr rrcg1cg2match1match2s r+r r s[ G  E 6??D S A S A S A dB MME (^^F Qq$5 ) )C Qq$F + +C WWSS51"a.4!Q-@@ A AF Rc&kkQ.Rq"%%nUF&C&CCII&QQQ WWSa%!Q$A?? @ @F c&kkQ.u Hr,cttrfddfSgd}tttfst dttr@jjr4jjr"fdtjDnfddfSttrOjD]2}t|tr |-||z}3||t|z fSdS)Nr5z term must be CG, Add, Mul or PowcDg|]}jSrR)rr)rGrncgrs r+rYz_cg_list..s' = = =qbii "" = = =r,) rrrrNotImplementedErrorrrEr]r*rr)rcoeffrHrs` @r+_cg_listrs($w1} B E dS#J ' 'F!"DEEE$!!3! 8  ! = = = = =E$(OO = = = = =7Aq= $+9  C#r""  # 5%E *** ++r,r)7r~sympy.concrete.summationsrsympy.core.addrsympy.core.exprrsympy.core.functionrsympy.core.mulrsympy.core.powerrsympy.core.relationalr sympy.core.singletonr sympy.core.symbolr r sympy.core.sympifyr (sympy.functions.elementary.miscellaneousr$sympy.functions.elementary.piecewiser sympy.printing.pretty.stringpictrr(sympy.functions.special.tensor_functionsrsympy.physics.wignerrrrrsympy.printing.precedencer__all__rrrrrrrrrrrrrrr rrRr,r+r*s #")))))) &&&&&& $$$$$$""""""--------&&&&&&999999::::::CCCCCCCCCCCCCCPPPPPPPPPPPP000000   xOxOxOxOxOtxOxOxOvS9S9S9S9S9S9S9S9lVPVPVPVPVPtVPVPVPremememememtemememP***Z+,+,+,\nnnnnn*/*/*/ZH!H!H!V                (+++++r,