U Zʹj[@sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZmZdd lmZdd lmZdd lmZddlmZmZddlmZddlm Z m!Z!m"Z"m#Z#ddl$m%Z%dddddgZ&GdddeZ'Gddde'Z(GdddeZ)GdddeZ*ddZ+ddZ,ddZ-d d!Z.d"d#Z/d$d%Z0d3d'd(Z1d)d*Z2d+d,Z3d-d.Z4d/d0Z5d1d2Z6d&S)4zClebsch-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) PRECEDENCECGWigner3jWigner6jWigner9jcg_simpc@seZdZdZdZddZeddZeddZed d Z ed d Z ed dZ eddZ eddZ ddZddZddZdS)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. TcCs&tt||||||f}tj|f|SNmapr r__new__)clsj1m1j2m2j3m3argsr(W/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/physics/quantum/cg.pyrQszWigner3j.__new__cCs |jdSNrr'selfr(r(r)r!Usz Wigner3j.j1cCs |jdSNr+r,r(r(r)r"Ysz Wigner3j.m1cCs |jdSNr+r,r(r(r)r#]sz Wigner3j.j2cCs |jdSNr+r,r(r(r)r$asz Wigner3j.m2cCs |jdSNr+r,r(r(r)r%esz Wigner3j.j3cCs |jdSNr+r,r(r(r)r&isz Wigner3j.m3cCstdd|jD S)Ncss|] }|jVqdSr is_number.0argr(r(r) osz'Wigner3j.is_symbolic..allr'r,r(r(r) is_symbolicmszWigner3j.is_symboliccsv||j||jf||j||jf||j||jffd}d}dgd}tdD]$tfddtdD|<q`d}tdD]}d}tdD]|} || } | d} | | } t | d| } t | d| } |dkr| }qt | d|}t | | }q|dkr8|}qt|D]} t | d}q@t | |}qt |}|S)Nr1r/r3c3s|]}|VqdSrwidthr;ijmr(r)r=zsz#Wigner3j._pretty.. )_printr!r"r#r$r%r&rangemaxrCrrightleftbelowparensr-printerr'ZhsepZvsepZmaxwDrEZD_rowsZwdeltaZwleftZwright_r(rFr)_prettyrs@  "       zWigner3j._prettycGs0t|j|j|j|j|j|j|jf}dt|S)NzH\left(\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right)) rrJr!r#r%r"r$r&tupler-rRr'labelr(r(r)_latexszWigner3j._latexcKs,|jrtdt|j|j|j|j|j|jSNzCoefficients must be numerical) r@ ValueErrorrr!r#r%r"r$r&r-hintsr(r(r)doitsz Wigner3j.doitN)__name__ __module__ __qualname____doc__Zis_commutativerpropertyr!r"r#r$r%r&r@rVrZr_r(r(r(r)r&s((       #c@s4eZdZdZeddZddZddZdd Zd S) 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). rr/cKs,|jrtdt|j|j|j|j|j|jSr[) r@r\rr!r#r%r"r$r&r]r(r(r)r_szCG.doitcGs|j|j|j|j|jfdd}|j|j|jfdd}t||}t | d}t | d}||kst | d||}||kst | d||}t dd|}t | |}t ||}|S)N,) delimiterrIC)Z _print_seqr!r"r#r$r%r&rLrCrrNrMrrOZabove)r-rRr'ZbottoppadrTr(r(r)rVs   z CG._prettycGs0t|j|j|j|j|j|j|jf}dt|S)NzC^{%s,%s}_{%s,%s,%s,%s}) rrJr%r&r!r"r#r$rWrXr(r(r)rZs z CG._latexN) r`rarbrcr precedencer_rVrZr(r(r(r)rs 6 c@seZdZdZddZeddZeddZedd Zed d Z ed d Z eddZ eddZ ddZ ddZddZdS)rzaClass for the Wigner-6j symbols See Also ======== Wigner3j: Wigner-3j symbols cCs&tt||||||f}tj|f|Srr)r r!r#j12r%rGj23r'r(r(r)rszWigner6j.__new__cCs |jdSr*r+r,r(r(r)r!sz Wigner6j.j1cCs |jdSr.r+r,r(r(r)r#sz Wigner6j.j2cCs |jdSr0r+r,r(r(r)rk sz Wigner6j.j12cCs |jdSr2r+r,r(r(r)r%sz Wigner6j.j3cCs |jdSr4r+r,r(r(r)rGsz Wigner6j.jcCs |jdSr6r+r,r(r(r)rlsz Wigner6j.j23cCstdd|jD S)Ncss|] }|jVqdSrr8r:r(r(r)r=sz'Wigner6j.is_symbolic..r>r,r(r(r)r@szWigner6j.is_symboliccs|||j||jf||j||jf||j||jffd}d}dgd}tdD]$tfddtdD|<q`d}tdD]}d}tdD]|} || } | d} | | } t | d| } t | d| } |dkr| }qt | d|}t | | }q|dkr8|}qt|D]} t | d}q@t | |}qt |jdd d }|S) Nr1r/rAr3c3s|]}|VqdSrrBrDrFr(r)r=)sz#Wigner6j._pretty..rI{}rNrM)rJr!r%r#rGrkrlrKrLrCrrMrNrOrPrQr(rFr)rV!s@  "      zWigner6j._prettycGs0t|j|j|j|j|j|j|jf}dt|S)NzJ\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right\}) rrJr!r#rkr%rGrlrWrXr(r(r)rZDszWigner6j._latexcKs,|jrtdt|j|j|j|j|j|jSr[) r@r\rr!r#rkr%rGrlr]r(r(r)r_Jsz Wigner6j.doitN)r`rarbrcrrdr!r#rkr%rGrlr@rVrZr_r(r(r(r)rs&       #c@seZdZdZddZeddZeddZedd Zed d Z ed d Z eddZ eddZ eddZ eddZeddZddZddZddZdS)rzaClass for the Wigner-9j symbols See Also ======== Wigner3j: Wigner-3j symbols c Cs,tt||||||||| f } tj|f| Srr) r r!r#rkr%j4j34j13j24rGr'r(r(r)rYszWigner9j.__new__cCs |jdSr*r+r,r(r(r)r!]sz Wigner9j.j1cCs |jdSr.r+r,r(r(r)r#asz Wigner9j.j2cCs |jdSr0r+r,r(r(r)rkesz Wigner9j.j12cCs |jdSr2r+r,r(r(r)r%isz Wigner9j.j3cCs |jdSr4r+r,r(r(r)rpmsz Wigner9j.j4cCs |jdSr6r+r,r(r(r)rqqsz Wigner9j.j34cCs |jdS)Nr+r,r(r(r)rrusz Wigner9j.j13cCs |jdS)Nr+r,r(r(r)rsysz Wigner9j.j24cCs |jdS)Nr+r,r(r(r)rG}sz Wigner9j.jcCstdd|jD S)Ncss|] }|jVqdSrr8r:r(r(r)r=sz'Wigner9j.is_symbolic..r>r,r(r(r)r@szWigner9j.is_symboliccs||j||j||jf||j||j||jf||j||j||j ffd}d}dgd}t dD]$t fddt dD|<q~d}t dD]}d}t dD]|} || } | d} | | } t | d| } t | d| } |dkr$| }qt |d|}t || }q|dkrV|}qt |D]} t |d}q^t ||}qt |jdd d }|S) Nr1r/rAr3c3s|]}|VqdSrrBrDrFr(r)r=sz#Wigner9j._pretty..rIrmrnro)rJr!r%rrr#rprsrkrqrGrKrLrCrrMrNrOrPrQr(rFr)rVsT     "      zWigner9j._prettyc Gs<t|j|j|j|j|j|j|j|j|j |j f }dt |S)NzZ\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \\ %s & %s & %s \end{array}\right\}) rrJr!r#rkr%rprqrrrsrGrWrXr(r(r)rZszWigner9j._latexc Ks8|jrtdt|j|j|j|j|j|j|j |j |j Sr[) r@r\rr!r#rkr%rprqrrrsrGr]r(r(r)r_sz Wigner9j.doitN)r`rarbrcrrdr!r#rkr%rprqrrrsrGr@rVrZr_r(r(r(r)rPs2          &cCsft|trt|St|tr$t|St|trBtdd|jDSt|tr^tt|j |j S|SdS)aSimplify and combine CG coefficients. Explanation =========== This function uses various symmetry and properties of sums and products of Clebsch-Gordan coefficients to simplify statements involving these terms [1]_. Examples ======== Simplify the sum over CG(a,alpha,0,0,a,alpha) for all alpha to 2*a+1 >>> from sympy.physics.quantum.cg import CG, cg_simp >>> a = CG(1,1,0,0,1,1) >>> b = CG(1,0,0,0,1,0) >>> c = CG(1,-1,0,0,1,-1) >>> cg_simp(a+b+c) 3 See Also ======== CG: Clebsh-Gordan coefficients References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. cSsg|] }t|qSr()rr:r(r(r) szcg_simp..N) isinstancer _cg_simp_addr _cg_simp_sumrr'rrbaseexper(r(r)rs!    cCsg}g}t|}|jD]}|trt|tr>|t|qt|trd}|jD]$}t|trn|t|9}qR||9}qR|tr||q||q||q||qt |\}}||t |\}}||t |\}}||t |t |S)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 r/) rr'hasrrxrappendrzr_check_varsh_871_1_check_varsh_871_2_check_varsh_872_9r)r~Zcg_part other_partr<Ztermstermotherr(r(r)rys2                   ryc Csttd\}}}}|t|||d||}d|dt|d}|t|}d|d}||} t|||||||||f||f|| S)N)aalphabltrr1r/)rr rrabs_check_cg_simp) term_listrrrrexprsimpsign build_expr index_exprr(r(r)rs  rc Csttd\}}}}|t|||| |d}td|dt|d}d|||t|}d|d}||} t|||||||||f||f|| S)N)rrcrrr1r/rA)rr rr rrr) rrrrrrrrrrr(r(r)rs rcCsttd\ }}}}}}}}} | t||||||d} tj} | t| } t||} t||}||dt| | |kfdt| |f||| kf}|||}t| | | | |||||||| f||||f|| \}}t||} ||}|d| | |d}|| | |||}t| | | | |||||||| f||||f|| \}}t||||||t||||||} t ||t ||} tj} t||} t||}||dt| | |kfdt| |f||| kf}|||}t| | | tj|||||||||f||||||f|| \}}t||} ||}|d| | |d}|| | |||}t| | | tj|||||||||f||||||f|| \}}||||fS)N) rralphaprbetabetaprgammarr1r/r) rr rr Onerrrrr)rrrrrrrrrrrrrxyrrZother1Zother2Zother3Zother4r(r(r)r)s8   2 2 2$  2 : :rc  sd} d} | tkrt| |t|dkr<| d7} q|jsR| d7} qfdd|D} dg|} t| tD]} t| || t|t|||| fd}|dkrq|| |jsq| ||d| ||||| |f| || |<qtdd| Dstd d| D}d d| D}||fd d|D| D]:}t |d |krx |d ||d |dqx| |||7} q| d7} q| 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 rNr/csg|]}||fqSr(r()r;r)sub_1r(r)rwsz"_check_cg_simp..)rcss|]}|dkVqdSrr(rDr(r(r)r=sz!_check_cg_simp..cSsg|]}t|dqS)r1)rr;rr(r(r)rwscSsg|] }|dqS)rr(rr(r(r)rwscsg|]}|qSr()pop)r;rG)rr(r)rwsr1r3) len _check_cgsubsr9rKanyminsortreverserr)rrrrr variablesZ dep_variablesZbuild_index_exprrrrEZsub_depZcg_indexrGZsub_2Zmin_ltindicesrr()rrr)rVs>) 6D& rNcCs^||}|dkrdS|dk rJt|ts0td|d|d|ksJdSt||krZ|SdS)z2Checks whether a term matches the given expressionNzsign must be a tuplerr/)matchrxrW TypeErrorrr)Zcg_termrlengthrmatchesr(r(r)rs   rcCst|}t|}t|}|Sr)_check_varsh_sum_871_1_check_varsh_sum_871_2_check_varsh_sum_872_4r}r(r(r)rzsrzc Csrtd}td}td}|tt|||d|||| |f}|dk rnt|dkrnd|dt|d|S|S)Nrrrrr1r/)r r rrrrrr)r~rrrrr(r(r)rs&rc Cstd}td}td}|td||t|||| |d|| |f}|dk rt|dkrtd|dt|d|S|S)NrrrrArr1r/) r r rrrrr rr)r~rrrrr(r(r)rs, rc Cstd}td}td}td}td}td}td}td}t||||||} t||||||} |t| | || |f|| |f} | dk rt| d krt||t||| S|t| d || |f|| |f} | dk rt| d krtj S|S) Nrrrrrcprgammaprtr1r5) r r rrrrrrr r) r~rrrrrrrrZcg1Zcg2Zmatch1Zmatch2r(r(r)rs"&&rcsttrfddfSgd}tttfs4tdttrvjjrvjjrjfddtjDn fddfSttrjD]"}t|tr |q||9}q||t |fSdS)Nr/z term must be CG, Add, Mul or Powcsg|]}jqSr()rr{)r;rUZcgrr(r)rwsz_cg_list..) rxrrrNotImplementedErrorr|r9rKr'rr)rZcoeffr<r(rr)_cg_lists         r)N)7rcZsympy.concrete.summationsrZsympy.core.addrZsympy.core.exprrZsympy.core.functionrZsympy.core.mulrZsympy.core.powerrZsympy.core.relationalrZsympy.core.singletonr Zsympy.core.symbolr r Zsympy.core.sympifyr Z(sympy.functions.elementary.miscellaneousr Z$sympy.functions.elementary.piecewiserZ sympy.printing.pretty.stringpictrrZ(sympy.functions.special.tensor_functionsrZsympy.physics.wignerrrrrZsympy.printing.precedencer__all__rrrrrryrrrrrrzrrrrr(r(r(r)sL              {VYh-.  -K