o 8VaY@s*dZddlmZmZmZmZmZmZmZm Z m Z m Z m Z m Z ddlmZmZddlmZddlmZmZmZmZddlmZgdZGdd d eZGd d d eZGd d d eZGdddeZddZddZ ddZ!ddZ"ddZ#ddZ$d)ddZ%dd Z&d!d"Z'd#d$Z(d%d&Z)d'd(Z*dS)*zClebsch-Gordon Coefficients.) AddexpandEqExprMul PiecewisePowsqrtSumsymbolssympifyWild) 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(:/usr/lib/python3/dist-packages/sympy/physics/quantum/cg.pyrGzWigner3j.__new__cC |jdSNrr'selfr(r(r)r!K z Wigner3j.j1cCr+Nr-r.r(r(r)r"Or0z Wigner3j.m1cCr+Nr-r.r(r(r)r#Sr0z Wigner3j.j2cCr+Nr-r.r(r(r)r$Wr0z Wigner3j.m2cCr+Nr-r.r(r(r)r%[r0z Wigner3j.j3cCr+Nr-r.r(r(r)r&_r0z Wigner3j.m3cCtdd|jD S)NcSg|]}|jqSr( is_number.0argr(r(r) ez(Wigner3j.is_symbolic..allr'r.r(r(r) is_symbolicczWigner3j.is_symboliccsp||j||jf||j||jf||j||jffd}d}dgd}tdD]tfddtdD|<q0d}tdD]f}d}tdD]A|} || } | d} | | } t | d| } t | d| } |dur| }qQt | d|}t | | }qQ|dur|}qIt|D] } t | d}qt | |}qIt |}|S)Nr4r2r6cg|] }|qSr(widthr@ijmr(r)rBpz$Wigner3j._pretty.. )_printr!r"r#r$r%r&rangemaxrKrrightleftbelowparensr/printerr'ZhsepZvsepZmaxwDrMZD_rowsZwdeltaZwleftZwright_r(rNr)_prettyhs@  "     zWigner3j._prettycG0t|j|j|j|j|j|j|jf}dt|S)NzH\left(\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right)) rrSr!r#r%r"r$r&tupler/r[r'Zlabelr(r(r)_latex  zWigner3j._latexcK,|jrtdt|j|j|j|j|j|jSNzCoefficients must be numerical) rF ValueErrorrr!r#r%r"r$r&r/Zhintsr(r(r)doitz Wigner3j.doitN)__name__ __module__ __qualname____doc__Zis_commutativerpropertyr!r"r#r$r%r&rFr_rcrir(r(r(r)rs*(       # 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). rr2cKrerf) rFrgrr!r#r%r"r$r&rhr(r(r)rirjzCG.doitcGs|j|j|j|j|jfdd}|j|j|jfdd}t||}t | d}t | d}||ksDt | d||}||ksWt | d||}t dd|}t | |}t ||}|S)N,)Z delimiterrRC)Z _print_seqr!r"r#r$r%r&rUrKrrWrVrrXZabove)r/r[r'ZbottopZpadr]r(r(r)r_s  z CG._prettycGr`)NzC^{%s,%s}_{%s,%s,%s,%s}) rrSr%r&r!r"r#r$rarbr(r(r)rcs  z CG._latexN) rkrlrmrnrZ precedencerir_rcr(r(r(r)rs  6 rc@seZdZdZddZeddZeddZedd Zed d Z ed d Z eddZ eddZ ddZ ddZddZdS)rzaClass for the Wigner-6j symbols See Also ======== Wigner3j: Wigner-3j symbols cCrrr)r r!r#j12r%rOj23r'r(r(r)rr*zWigner6j.__new__cCr+r,r-r.r(r(r)r!r0z Wigner6j.j1cCr+r1r-r.r(r(r)r#r0z Wigner6j.j2cCr+r3r-r.r(r(r)rsr0z Wigner6j.j12cCr+r5r-r.r(r(r)r%r0z Wigner6j.j3cCr+r7r-r.r(r(r)rO r0z Wigner6j.jcCr+r9r-r.r(r(r)rtr0z Wigner6j.j23cCr;)NcSr<r(r=r?r(r(r)rBrCz(Wigner6j.is_symbolic..rDr.r(r(r)rFrGzWigner6j.is_symboliccsv||j||jf||j||jf||j||jffd}d}dgd}tdD]tfddtdD|<q0d}tdD]f}d}tdD]A|} || } | d} | | } t | d| } t | d| } |dur| }qQt | d|}t | | }qQ|dur|}qIt|D] } t | d}qt | |}qIt |jdd d }|S) Nr4r2rHr6crIr(rJrLrNr(r)rBrQz$Wigner6j._pretty..rR{}rWrV)rSr!r%r#rOrsrtrTrUrKrrVrWrXrYrZr(rNr)r_s@  "    zWigner6j._prettycGr`)NzJ\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right\}) rrSr!r#rsr%rOrtrarbr(r(r)rc:rdzWigner6j._latexcKrerf) rFrgrr!r#rsr%rOrtrhr(r(r)ri@rjz Wigner6j.doitN)rkrlrmrnrror!r#rsr%rOrtrFr_rcrir(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#rsr%j4j34j13j24rOr'r(r(r)rOszWigner9j.__new__cCr+r,r-r.r(r(r)r!Sr0z Wigner9j.j1cCr+r1r-r.r(r(r)r#Wr0z Wigner9j.j2cCr+r3r-r.r(r(r)rs[r0z Wigner9j.j12cCr+r5r-r.r(r(r)r%_r0z Wigner9j.j3cCr+r7r-r.r(r(r)rxcr0z Wigner9j.j4cCr+r9r-r.r(r(r)rygr0z Wigner9j.j34cCr+)Nr-r.r(r(r)rzkr0z Wigner9j.j13cCr+)Nr-r.r(r(r)r{or0z Wigner9j.j24cCr+)Nr-r.r(r(r)rOsr0z Wigner9j.jcCr;)NcSr<r(r=r?r(r(r)rByrCz(Wigner9j.is_symbolic..rDr.r(r(r)rFwrGzWigner9j.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) Nr4r2rHr6crIr(rJrLrNr(r)rBrQz$Wigner9j._pretty..rRrurvrw)rSr!r%rzr#rxr{rsryrOrTrUrKrrVrWrXrYrZr(rNr)r_|sP   "    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\}) rrSr!r#rsr%rxryrzr{rOrarbr(r(r)rcs zWigner9j._latexc Ks8|jrtdt|j|j|j|j|j|j|j |j |j Srf) rFrgrr!r#rsr%rxryrzr{rOrhr(r(r)ris*z Wigner9j.doitN)rkrlrmrnrror!r#rsr%rxryrzr{rOrFr_rcrir(r(r(r)rFs4          & 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)rBzcg_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 r2) rr'Zhasrrr appendrr_check_varsh_871_1_check_varsh_871_2_check_varsh_872_9r)rZcg_part other_partrAZtermstermotherr(r(r)rs2                   rc Csttd\}}}}|t|||d||}d|dt|d}|t|}d|d}||} t|||||||||f||f|| S)N)aalphabltrr4r2)rr rrabs_check_cg_simp) term_listrrrrexprsimpsign build_expr index_exprr(r(r)r s  $rc Csttd\}}}}|t|||| |d}td|dt|d}d|||t|}d|d}||} t|||||||||f||f|| S)N)rrcrrr4r2rH)rr rr rrr) rrrrrrrrrrr(r(r)rs $rcCsttd\ }}}}}}}}} | t||||||d} d} | t| } t||} t||}||dt| | |kfdt| |f||| kf}|||}t| | | | |||||||| f||||f|| \}}t||} ||}|d| | |d}|| | |||}t| | | | |||||||| f||||f|| \}}t||||||t||||||} t||t||} td} t||} t||}||dt| | |kfdt| |f||| kf}|||}t| | | td|||||||||f||||||f|| \}}t||} ||}|d| | |d}|| | |||}t| | | td|||||||||f||||||f|| \}}||||fS)N) rralphaprbetabetaprgammarr4r2r) rr rrrrrrr )rrrrrrrrrrrrrxyrrZother1Zother2Zother3Zother4r(r(r)rs8   2 2 2$  2 < <rc  sd} d} | tkrt| |t|dur| d7} qt|js*| d7} qfdd|D} dg|} t| tD]O} t| || t|t|||| fd}|durdqBt|| |jspqB| ||d| ||||| |f| || |<qBtdd| Drtd d| D}d d| D}|| fd d|D| D]}t |d |krو |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 rNr2csg|]}||fqSr(r()r@r)sub_1r(r)rBz"_check_cg_simp..)rcss|]}|duVqdSrr(rLr(r(r) sz!_check_cg_simp..cSsg|]}t|dqS)r4)rr@rr(r(r)rBrcSsg|]}|dqS)rr(rr(r(r)rBrcsg|]}|qSr()pop)r@rO)rr(r)rBsr4r6) len _check_cgr subsr>rTrEminsortreverserr)rrrrrZ variablesZ dep_variablesZbuild_index_exprrrrMZsub_depZcg_indexrOZsub_2Zmin_ltindicesrr()rrr)rLsB) 6D" rNcCs^||}|dur dS|dur%t|tstd|d|d|ks%dSt||kr-|SdS)z2Checks whether a term matches the given expressionNzsign must be a tuplerr2)matchrra TypeErrorrr)Zcg_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)Nrrrrr4r2)r r rr rrrr)rrrrrr(r(r)rs&rc Cstd}td}td}|td||t|||| |d|| |f}|dur@t|dkr@td|dt|d|S|S)NrrrrHrr4r2) r r rr rrr 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{d S|S) Nrrrrrcprgammapr|r4r8r2)r r rrr rrr) rrrrrrrrrZcg1Zcg2Zmatch1Zmatch2r(r(r)rs"&&rcsttr fddfSgd}ttsttstdttrBtjjrBtjjr<fddtjDnfddfSttrej D]}t|trW |qJ||9}qJ||t |fSdS)Nr2z term must be CG, Add, Mul or Powcsg|]}jqSr()rr)r@r^Zcgrr(r)rBrz_cg_list..) rrrrNotImplementedErrorr rr>rTr'rr)rZcoeffrAr(rr)_cg_lists"         rr)+rnZsympyrrrrrrrr r r r r Z sympy.printing.pretty.stringpictrrZ(sympy.functions.special.tensor_functionsrZsympy.physics.wignerrrrrZsympy.printing.precedencer__all__rrrrrrrrrrrrrrrrr(r(r(r)s.8   {VYh-.  - K