Otj[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 V/var/www/html/CA-Chatbot/venv/lib/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$:: :::::::r,c^||j||jf||j||jf||j||jffd}d}dgdz}tdD].tfdtdD|</d}tdD]}d}tdD]|} || z } | dz} | | z } t| d| z} t| d| z} || }zt| d|z}t| | }||}t|D]} t| d}t| |}t|}|S)Nr8r5r;c3XK|]$}|V%dSrwidthrGijms r+rIz#Wigner3j._pretty..z3<'> ? ^^DG $ $gnnTW&=&= > ^^DG $ $gnnTW&=&= > @tAvq = =A<<<<<588<<<<> >$'47DGTWdgNNNr,N)__name__ __module__ __qualname____doc__is_commutativer"propertyr$r%r&r'r(r)rMrmrrrxr,r+rr&s%&&PN(((XXXXXX;;X;!!!F OOOOOr,rc:eZdZdZeddz ZdZdZdZdS)raClass for Clebsch-Gordan coefficient. Explanation =========== Clebsch-Gordan coefficients describe the angular momentum coupling between two systems. The coefficients give the expansion of a coupled total angular momentum state and an uncoupled tensor product state. The Clebsch-Gordan coefficients are defined as [1]_: .. math :: C^{j_3,m_3}_{j_1,m_1,j_2,m_2} = \left\langle j_1,m_1;j_2,m_2 | j_3,m_3\right\rangle Parameters ========== j1, m1, j2, m2 : Number, Symbol Angular momenta of states 1 and 2. j3, m3: Number, Symbol Total angular momentum of the coupled system. Examples ======== Define a Clebsch-Gordan coefficient and evaluate its value >>> from sympy.physics.quantum.cg import CG >>> from sympy import S >>> cg = CG(S(3)/2, S(3)/2, S(1)/2, -S(1)/2, 1, 1) >>> cg CG(3/2, 3/2, 1/2, -1/2, 1, 1) >>> cg.doit() sqrt(3)/2 >>> CG(j1=S(1)/2, m1=-S(1)/2, j2=S(1)/2, m2=+S(1)/2, j3=1, m3=0).doit() sqrt(2)/2 Compare [2]_. See Also ======== Wigner3j: Wigner-3j symbols References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. .. [2] `Clebsch-Gordan Coefficients, Spherical Harmonics, and d Functions `_ in P.A. Zyla *et al.* (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020). rr5c |jrtdt|j|j|j|j|j|jSrt) rMrurr$r&r(r%r'r)rvs r+rxzCG.doits@   ?=>> >dgtw$'47SSSr,cL||j|j|j|jfd}||j|jfd}t||}t| d}t| d}||ks4t| d||z z}||ks4t| d||z z}tdd|zz}t| |}t||}|S)N,) delimiterrYC) _print_seqr$r%r&r'r(r)r\rSrr^r]rr_above)r1rbr*bottoppadrhs r+rmz CG._prettysT  Wdgtw 0C!AA  $'47!3s CC#))++syy{{++#((3--(#((3--(ciikk!!ciiS399;;->(?@@ACciikk!!ciiS399;;->(?@@AC sSW} % %  %  %r,c t|j|j|j|j|j|j|jf}dt|zS)NzC^{%s,%s}_{%s,%s,%s,%s}) r!rZr(r)r$r%r&r'rorps r+rrz CG._latexsDGNTWdgtwGTWdg%/00)E%LL88r,N) ryrzr{r|r precedencerxrmrrrr,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(rVj23r*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+rVz Wigner6j.jr2r,c|jdSr@r/r0s r+rz Wigner6j.j23r2r,c@td|jD S)Nc3$K|] }|jV dSrrDrFs r+rIz'Wigner6j.is_symbolic..rJr,rKr0s r+rMzWigner6j.is_symbolicrNr,cd||j||jf||j||jf||j||jffd}d}dgdz}tdD].tfdtdD|</d}tdD]}d}tdD]|} || z } | dz} | | z } t| d| z} t| d| z} || }zt| d|z}t| | }||}t|D]} t| d}t| |}t|dd }|S) Nr8r5rPr;c3XK|]$}|V%dSrrRrTs r+rIz#Wigner6j._pretty..)rXr,rY{}r^r])rZr$r(r&rVrrr[r\rSrr]r^r_r`ras @@r+rmzWigner6j._pretty!snnTW%%w~~dg'>'> ? ^^DG $ $gnnTV&<&< = ^^DH % %w~~dh'?'? @ BtAvq = =A<<<<<588<<<<> >$'48TWdfdhOOOr,N)ryrzr{r|r"r~r$r&rr(rVrrMrmrrrxrr,r+rrs(((XXXXXX;;X;!!!F PPPPPr,rceZdZdZdZedZedZedZedZ edZ edZ ed Z ed Z ed Zed Zd ZdZdZdS)rzaClass for the Wigner-9j symbols See Also ======== Wigner3j: Wigner-3j symbols c btt||||||||| f } tj|g| RSrr ) r#r$r&rr(j4j34j13j24rVr*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+rVz Wigner9j.j}r2r,c@td|jD S)Nc3$K|] }|jV dSrrDrFs r+rIz'Wigner9j.is_symbolic..rJr,rKr0s r+rMzWigner9j.is_symbolicrNr,c||j||j||jf||j||j||jf||j||j||j ffd}d}dgdz}tdD].tfdtdD|</d}tdD]}d}tdD]|} || z } | dz} | | z } t| d| z} t| d| z} || }zt|d|z}t|| }||}t|D]} t|d}t||}t|dd }|S) Nr8r5rPr;c3XK|]$}|V%dSrrRrTs r+rIz#Wigner9j._pretty..rXr,rYrrr)rZr$r(rr&rrrrrVr[r\rSrr]r^r_r`ras @@r+rmzWigner9j._prettys` ^^!..117>>$(3K3K M ^^!..117>>$(3K3K M ^^DH % %w~~dh'?'?PTPVAWAW X  Z tAvq = =A<<<<<588<<<<> >$'48TWdgtxQUQY[_[ceieklllr,N)ryrzr{r|r"r~r$r&rr(rrrrrVrMrmrrrxrr,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|Sr)rrFs r+ zcg_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. 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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 Srr)rGrsub_1s r+rz"_check_cg_simp..s!888QAuQx=888r,)rc3K|]}|duV dSrr)rGrUs r+rIz!_check_cg_simp..s&//19//////r,c8g|]}t|dS)r8)rrGrs r+rz"_check_cg_simp..s"???TCQLL???r,cg|] }|d S)rrrs r+rz"_check_cg_simp..s555DQ555r,c:g|]}|Sr)pop)rGrVrs r+rz"_check_cg_simp..s% 1 1 11immA 1 1 1r,r8r;) len _check_cgsubsrEr[anyminsortreverserr)rrrrr variables dep_variablesbuild_index_exprrrrUsub_depcg_indexrVsub_2min_ltindicesrrs ` @r+rrVsZRJ A c)nn  )A,c)nn== = FA $$U++5  FA 8888-8886*//666q#i..)) 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Hr,c td}td}td}td}td}td}td}td}t||||||} t||||||} |t | | z|| |f|| |f} | Gt | d kr4t ||t ||z| S|t | d z|| |f|| |f} | t | d kr tj S|S) Nrrrrrcprgammaprr8r>) r r rrrrrrr r) rrrrrrr rrcg1cg2match1match2s r+r r sQ 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 c&kkQ..q"%%nUF&C&CCII&QQQ WWSa%!Q$A?? @ @F c&kkQ..u Hr,cttrfddfSgd}tttfst dttr@jjr4jjr"fdtjDnfddfSttrOjD]2}t|tr |-||z}3||t|z fSdS)Nr5z term must be CG, Add, Mul or PowcDg|]}jSr)rr)rGrlcgrs r+rz_cg_list..s' = = =qbii "" = = =r,) rrrrNotImplementedErrorrrEr[r*rr)rcoeffrHrs` @r+_cg_listrs($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 rrr,r+r*s #")))))) &&&&&& $$$$$$""""""--------&&&&&&999999::::::CCCCCCCCCCCCCCPPPPPPPPPPPP000000   xOxOxOxOxOtxOxOxOvS9S9S9S9S9S9S9S9lVPVPVPVPVPtVPVPVPremememememtemememP***Z+,+,+,\nnnnnn*/*/*/ZH!H!H!V                (+++++r,