o jg[@sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZmZdd lmZdd lmZdd lmZddlmZmZddlmZddlm Z m!Z!m"Z"m#Z#ddl$m%Z%gdZ&GdddeZ'Gddde'Z(GdddeZ)GdddeZ*ddZ+ddZ,dd Z-d!d"Z.d#d$Z/d%d&Z0d4d(d)Z1d*d+Z2d,d-Z3d.d/Z4d0d1Z5d2d3Z6d'S)5zClebsch-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@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. TcC(tt||||||f}tj|g|RSNmapr r__new__)clsj1m1j2m2j3m3argsr)P/var/www/html/zoom/venv/lib/python3.10/site-packages/sympy/physics/quantum/cg.pyr QzWigner3j.__new__cC |jdSNrr(selfr)r)r*r"U z Wigner3j.j1cCr,Nr.r/r)r)r*r#Yr1z Wigner3j.m1cCr,Nr.r/r)r)r*r$]r1z Wigner3j.j2cCr,Nr.r/r)r)r*r%ar1z Wigner3j.m2cCr,Nr.r/r)r)r*r&er1z Wigner3j.j3cCr,Nr.r/r)r)r*r'ir1z Wigner3j.m3cCtdd|jD S)Ncs|]}|jVqdSr is_number.0argr)r)r* oz'Wigner3j.is_symbolic..allr(r/r)r)r* is_symbolicmzWigner3j.is_symboliccsp||j||jf||j||jf||j||jffd}d}dgd}tdD]tfddtdD|<q0d}tdD]f}d}tdD]A|} || } | d} | | } t | d| } t | d| } |dur| }qQt | d|}t | | }qQ|dur|}qIt|D] } t | d}qt | |}qIt |}|S)Nr5r3r7c3 |] }|VqdSrwidthrAijmr)r*rCzz#Wigner3j._pretty.. )_printr"r#r$r%r&r'rangemaxrLrrightleftbelowparensr0printerr(hsepvsepmaxwDrND_rowswdeltawleftwright_r)rOr*_prettyrs@  "     zWigner3j._prettycG0t|j|j|j|j|j|j|jf}dt|S)NzH\left(\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right)) rrTr"r$r&r#r%r'tupler0r\r(labelr)r)r*_latex  zWigner3j._latexcK,|jrtdt|j|j|j|j|j|jSNzCoefficients must be numerical) rG ValueErrorrr"r$r&r#r%r'r0hintsr)r)r*doitz Wigner3j.doitN)__name__ __module__ __qualname____doc__is_commutativer propertyr"r#r$r%r&r'rGrgrlrsr)r)r)r*r&s*(       # rc@s4eZdZdZeddZddZddZdd Zd S) raClass for Clebsch-Gordan coefficient. Explanation =========== Clebsch-Gordan coefficients describe the angular momentum coupling between two systems. The coefficients give the expansion of a coupled total angular momentum state and an uncoupled tensor product state. The Clebsch-Gordan coefficients are defined as [1]_: .. math :: C^{j_3,m_3}_{j_1,m_1,j_2,m_2} = \left\langle j_1,m_1;j_2,m_2 | j_3,m_3\right\rangle Parameters ========== j1, m1, j2, m2 : Number, Symbol Angular momenta of states 1 and 2. j3, m3: Number, Symbol Total angular momentum of the coupled system. Examples ======== Define a Clebsch-Gordan coefficient and evaluate its value >>> from sympy.physics.quantum.cg import CG >>> from sympy import S >>> cg = CG(S(3)/2, S(3)/2, S(1)/2, -S(1)/2, 1, 1) >>> cg CG(3/2, 3/2, 1/2, -1/2, 1, 1) >>> cg.doit() sqrt(3)/2 >>> CG(j1=S(1)/2, m1=-S(1)/2, j2=S(1)/2, m2=+S(1)/2, j3=1, m3=0).doit() sqrt(2)/2 Compare [2]_. See Also ======== Wigner3j: Wigner-3j symbols References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. .. [2] `Clebsch-Gordan Coefficients, Spherical Harmonics, and d Functions `_ in P.A. Zyla *et al.* (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020). rr3cKrnro) rGrprr"r$r&r#r%r'rqr)r)r*rsrtzCG.doitcGs|j|j|j|j|jfdd}|j|j|jfdd}t||}t | d}t | d}||ksDt | d||}||ksWt | d||}t dd|}t | |}t ||}|S)N,) delimiterrSC) _print_seqr"r#r$r%r&r'rVrLrrXrWrrYabove)r0r\r(bottoppadrbr)r)r*rgs  z CG._prettycGrh)NzC^{%s,%s}_{%s,%s,%s,%s}) rrTr&r'r"r#r$r%rirjr)r)r*rls  z CG._latexN) rurvrwrxr precedencersrgrlr)r)r)r*rs  6 rc@seZdZdZddZeddZeddZedd Zed d Z ed d Z eddZ eddZ ddZ ddZddZdS)rzaClass for the Wigner-6j symbols See Also ======== Wigner3j: Wigner-3j symbols cCrrr)r!r"r$j12r&rPj23r(r)r)r*r r+zWigner6j.__new__cCr,r-r.r/r)r)r*r"r1z Wigner6j.j1cCr,r2r.r/r)r)r*r$r1z Wigner6j.j2cCr,r4r.r/r)r)r*r r1z Wigner6j.j12cCr,r6r.r/r)r)r*r&r1z Wigner6j.j3cCr,r8r.r/r)r)r*rPr1z Wigner6j.jcCr,r:r.r/r)r)r*rr1z Wigner6j.j23cCr<)Ncsr=rr>r@r)r)r*rCrDz'Wigner6j.is_symbolic..rEr/r)r)r*rGrHzWigner6j.is_symboliccsv||j||jf||j||jf||j||jffd}d}dgd}tdD]tfddtdD|<q0d}tdD]f}d}tdD]A|} || } | d} | | } t | d| } t | d| } |dur| }qQt | d|}t | | }qQ|dur|}qIt|D] } t | d}qt | |}qIt |jdd d }|S) Nr5r3rIr7c3rJrrKrMrOr)r*rC)rRz#Wigner6j._pretty..rS{}rXrW)rTr"r&r$rPrrrUrVrLrrWrXrYrZr[r)rOr*rg!s@  "    zWigner6j._prettycGrh)NzJ\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right\}) rrTr"r$rr&rPrrirjr)r)r*rlDrmzWigner6j._latexcKrnro) rGrprr"r$rr&rPrrqr)r)r*rsJrtz Wigner6j.doitN)rurvrwrxr rzr"r$rr&rPrrGrgrlrsr)r)r)r*rs(       # rc@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|g| RSrr) r!r"r$rr&j4j34j13j24rPr(r)r)r*r YszWigner9j.__new__cCr,r-r.r/r)r)r*r"]r1z Wigner9j.j1cCr,r2r.r/r)r)r*r$ar1z Wigner9j.j2cCr,r4r.r/r)r)r*rer1z Wigner9j.j12cCr,r6r.r/r)r)r*r&ir1z Wigner9j.j3cCr,r8r.r/r)r)r*rmr1z Wigner9j.j4cCr,r:r.r/r)r)r*rqr1z Wigner9j.j34cCr,)Nr.r/r)r)r*rur1z Wigner9j.j13cCr,)Nr.r/r)r)r*ryr1z Wigner9j.j24cCr,)Nr.r/r)r)r*rP}r1z Wigner9j.jcCr<)Ncsr=rr>r@r)r)r*rCrDz'Wigner9j.is_symbolic..rEr/r)r)r*rGrHzWigner9j.is_symboliccs||j||j||jf||j||j||jf||j||j||j ffd}d}dgd}t dD]t fddt dD|<q?d}t dD]f}d}t dD]A|} || } | d} | | } t | d| } t | d| } |dur| }q`t |d|}t || }q`|dur|}qXt |D] } t |d}qt ||}qXt |jdd d }|S) Nr5r3rIr7c3rJrrKrMrOr)r*rCrRz#Wigner9j._pretty..rSrrr)rTr"r&rr$rrrrrPrUrVrLrrWrXrYrZr[r)rOr*rgsP   "    zWigner9j._prettyc Gs<t|j|j|j|j|j|j|j|j|j |j f }dt |S)NzZ\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \\ %s & %s & %s \end{array}\right\}) rrTr"r$rr&rrrrrPrirjr)r)r*rls zWigner9j._latexc Ks8|jrtdt|j|j|j|j|j|j|j |j |j Sro) rGrprr"r$rr&rrrrrPrqr)r)r*rss*z Wigner9j.doitN)rurvrwrxr rzr"r$rr&rrrrrPrGrgrlrsr)r)r)r*rPs4          & rcCsbt|tr t|St|trt|St|tr!tdd|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. cSsg|]}t|qSr))rr@r)r)r* zcg_simp..) isinstancer _cg_simp_addr _cg_simp_sumrr(rrbaseexper)r)r*rs !   rcCsg}g}t|}|jD]M}|trSt|tr|t|q t|trMd}|jD]}t|tr7|t|9}q)||9}q)|trG||q ||q ||q ||q t |\}}||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 r3) rr(hasrrrappendrr_check_varsh_871_1_check_varsh_871_2_check_varsh_872_9r)rcg_part other_partrBtermstermotherr)r)r*rs2                   rc Csttd\}}}}|t|||d||}d|dt|d}|t|}d|d}||} t|||||||||f||f|| S)N)aalphabltrr5r3)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)rrcrrr5r3rI)rr rr rrr) rrrrrrrrrrr)r)r*rs $rcCsttd\ }}}}}}}}} | t||||||d} tj} | t| } t||} t||}||dt| | |kfdt| |f||| kf}|||}t| | | | |||||||| f||||f|| \}}t||} ||}|d| | |d}|| | |||}t| | | | |||||||| f||||f|| \}}t||||||t||||||} t ||t ||} tj} t||} t||}||dt| | |kfdt| |f||| kf}|||}t| | | tj|||||||||f||||||f|| \}}t||} ||}|d| | |d}|| | |||}t| | | tj|||||||||f||||||f|| \}}||||fS)N) rralphaprbetabetaprgammarr5r3r) rr rr Onerrrrr)rrrrrrrrrrrrrxyrrother1other2other3other4r)r)r*r)s8   2 2 2$  2 : :rc  sd} d} | tkrt| |t|dur| d7} q|js(| d7} qfdd|D} dg|} t| tD]M} t| || t|t|||| fd}|durbq@|| |jslq@| ||d| ||||| |f| || |<q@tdd| Dstd d| D}d d| D}||fd d|D| D]}t |d |krՈ |d ||d |dq| |||7} n| d7} | tks | 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 rNr3csg|]}||fqSr)r))rAr)sub_1r)r*rz"_check_cg_simp..)rcss|]}|duVqdSrr)rMr)r)r*rCsz!_check_cg_simp..cSsg|]}t|dqS)r5)rrArr)r)r*rrcSsg|]}|dqS)rr)rr)r)r*rrcsg|]}|qSr))pop)rArP)rr)r*rsr5r7) len _check_cgsubsr?rUanyminsortreverserr)rrrrr variables dep_variablesbuild_index_exprrrrNsub_depcg_indexrPsub_2min_ltindicesrr))rrr*rVsB)  6D" rNcCs^||}|dur dS|dur%t|tstd|d|d|ks%dSt||kr-|SdS)z2Checks whether a term matches the given expressionNzsign must be a tuplerr3)matchrri TypeErrorrr)cg_termrlengthrmatchesr)r)r*rs   rcCst|}t|}t|}|Sr)_check_varsh_sum_871_1_check_varsh_sum_871_2_check_varsh_sum_872_4rr)r)r*rsrc Csrtd}td}td}|tt|||d|||| |f}|dur7t|dkr7d|dt|d|S|S)Nrrrrr5r3)r r rrrrrr)rrrrrr)r)r*rs&rc Cstd}td}td}|td||t|||| |d|| |f}|dur@t|dkr@td|dt|d|S|S)NrrrrIrr5r3) r r rrrrr rr)rrrrrr)r)r*rs, rc Cstd}td}td}td}td}td}td}td}t||||||} t||||||} |t| | || |f|| |f} | dur\t| d kr\t||t||| S|t| d || |f|| |f} | dur|t| d kr|tj S|S) Nrrrrrcprgammaprr5r9) r r rrrrrrr r) rrrrrrrrrcg1cg2match1match2r)r)r*rs"&&rcsttr fddfSgd}tttfstdttr;jjr;jjr5fddtjDnfddfSttr^jD]}t|trP |qC||9}qC||t |fSdS)Nr3z term must be CG, Add, Mul or Powcsg|]}jqSr))rr)rArfcgrr)r*rrz_cg_list..) rrrrNotImplementedErrorrr?rUr(rr)rcoeffrBr)rr*_cg_lists"        rr)7rxsympy.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__rrrrrrrrrrrrrrrrr)r)r)r*sD              {VYh-.  - K