B [W@s@dZddlmZmZddlmZmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZddlmZmZddlmZddlmZmZmZmZdd d d d gZGd d d eZGdddeZGdd d eZ Gdd d eZ!dd Z"ddZ#ddZ$ddZ%ddZ&ddZ'd)ddZ(dd Z)d!d"Z*d#d$Z+d%d&Z,d'd(Z-dS)*zClebsch-Gordon Coefficients.)print_functiondivision) AddexpandEqExprMul PiecewisePowsqrtSumsymbolssympifyWild)range) prettyForm stringPict)KroneckerDelta)clebsch_gordan wigner_3j wigner_6j wigner_9jCGWigner3jWigner6jWigner9jcg_simpc@seZdZdZdZddZeddZeddZed d Z ed d Z ed dZ eddZ eddZ ddZddZddZdS)raClass for the Wigner-3j symbols 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|S)N)maprr__new__)clsj1m1j2m2j3m3argsr'7lib/python3.7/site-packages/sympy/physics/quantum/cg.pyrFszWigner3j.__new__cCs |jdS)Nr)r&)selfr'r'r(r Jsz Wigner3j.j1cCs |jdS)N)r&)r)r'r'r(r!Nsz Wigner3j.m1cCs |jdS)N)r&)r)r'r'r(r"Rsz Wigner3j.j2cCs |jdS)N)r&)r)r'r'r(r#Vsz Wigner3j.m2cCs |jdS)N)r&)r)r'r'r(r$Zsz Wigner3j.j3cCs |jdS)N)r&)r)r'r'r(r%^sz Wigner3j.m3cCstdd|jD S)NcSsg|] }|jqSr') is_number).0argr'r'r( dsz(Wigner3j.is_symbolic..)allr&)r)r'r'r( is_symbolicbszWigner3j.is_symboliccs||j||jf||j||jf||j||jffd}d}dgd}x0tdD]$tfddtdD|<qbWd}xtdD]}d}xtdD]|} || } | d} | | } t | d| } t | d| } |dkr| }qt | d|}t | | }qW|dkrB|}qx t|D]} t | d}qLWt | |}qWt |}|S)Nr+r*r,csg|]}|qSr')width)r0i)jmr'r(r2osz$Wigner3j._pretty.. )_printr r!r"r#r$r%rmaxr6rrightleftbelowparens)r)printerr&hsepvsepmaxwDr7D_rowswdeltawleftwright_r')r8r9r(_prettygs> $    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)) rr;r r"r$r!r#r%tuple)r)rAr&labelr'r'r(_latexszWigner3j._latexcKs,|jrtdt|j|j|j|j|j|jS)NzCoefficients must be numerical) r4 ValueErrorrr r"r$r!r#r%)r)hintsr'r'r(doitsz Wigner3j.doitN)__name__ __module__ __qualname____doc__Zis_commutativerpropertyr r!r"r#r$r%r4rLrOrRr'r'r'r(rs$       #c@s(eZdZdZddZddZddZdS) ra1Class for Clebsch-Gordan coefficient 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_1,m_1}_{j_2,m_2,j_3,m_3} = \langle j_1,m_1;j_2,m_2 | j_3,m_3\rangle Parameters ========== j1, m1, j2, m2, j3, m3 : Number, Symbol Terms determining the angular momentum of coupled angular momentum systems. 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 See Also ======== Wigner3j: Wigner-3j symbols References ========== .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. cKs,|jrtdt|j|j|j|j|j|jS)NzCoefficients must be numerical) r4rPrr r"r$r!r#r%)r)rQr'r'r(rRszCG.doitcGs|j|j|j|j|jfdd}|j|j|jfdd}t||}t | d}t | d}||kst | d||}||kst | d||}t dd|}t | |}t ||}|S)N,)Z delimiterr:C)Z _print_seqr r!r"r#r$r%r<r6rr>r=rr?Zabove)r)rAr&ZbottopZpadrGr'r'r(rLs  z CG._prettycGs0t|j|j|j|j|j|j|jf}dt|S)NzC^{%s,%s}_{%s,%s,%s,%s}) rr;r$r%r r!r"r#rM)r)rAr&rNr'r'r(rOsz CG._latexN)rSrTrUrVrRrLrOr'r'r'r(rs(c@seZdZdZddZeddZeddZedd Zed d Z ed d Z eddZ eddZ ddZ ddZddZdS)rzaClass for the Wigner-6j symbols See Also ======== Wigner3j: Wigner-3j symbols cCs&tt||||||f}tj|f|S)N)rrrr)rr r"j12r$r8j23r&r'r'r(rszWigner6j.__new__cCs |jdS)Nr)r&)r)r'r'r(r sz Wigner6j.j1cCs |jdS)Nr*)r&)r)r'r'r(r"sz Wigner6j.j2cCs |jdS)Nr+)r&)r)r'r'r(r[sz Wigner6j.j12cCs |jdS)Nr,)r&)r)r'r'r(r$sz Wigner6j.j3cCs |jdS)Nr-)r&)r)r'r'r(r8sz Wigner6j.jcCs |jdS)Nr.)r&)r)r'r'r(r\sz Wigner6j.j23cCstdd|jD S)NcSsg|] }|jqSr')r/)r0r1r'r'r(r2sz(Wigner6j.is_symbolic..)r3r&)r)r'r'r(r4szWigner6j.is_symboliccs||j||jf||j||jf||j||jffd}d}dgd}x0tdD]$tfddtdD|<qbWd}xtdD]}d}xtdD]|} || } | d} | | } t | d| } t | d| } |dkr| }qt | d|}t | | }qW|dkrB|}qx t|D]} t | d}qLWt | |}qWt |jdd d }|S) Nr+r*r5r,csg|]}|qSr')r6)r0r7)r8r9r'r(r2sz$Wigner6j._pretty..r:{})r>r=)r;r r$r"r8r[r\rr<r6rr=r>r?r@)r)rAr&rBrCrDrEr7rFrGrHrIrJrKr')r8r9r(rLs> $   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\}) rr;r r"r[r$r8r\rM)r)rAr&rNr'r'r(rO*szWigner6j._latexcKs,|jrtdt|j|j|j|j|j|jS)NzCoefficients must be numerical) r4rPrr r"r[r$r8r\)r)rQr'r'r(rR0sz Wigner6j.doitN)rSrTrUrVrrWr r"r[r$r8r\r4rLrOrRr'r'r'r(rs       #c@seZdZdZddZeddZeddZedd Zed d Z ed d Z eddZ eddZ eddZ eddZeddZddZddZddZdS)rzaClass for the Wigner-9j symbols See Also ======== Wigner3j: Wigner-3j symbols c Cs,tt||||||||| f } tj|f| S)N)rrrr) rr r"r[r$j4j34j13j24r8r&r'r'r(r?szWigner9j.__new__cCs |jdS)Nr)r&)r)r'r'r(r Csz Wigner9j.j1cCs |jdS)Nr*)r&)r)r'r'r(r"Gsz Wigner9j.j2cCs |jdS)Nr+)r&)r)r'r'r(r[Ksz Wigner9j.j12cCs |jdS)Nr,)r&)r)r'r'r(r$Osz Wigner9j.j3cCs |jdS)Nr-)r&)r)r'r'r(r_Ssz Wigner9j.j4cCs |jdS)Nr.)r&)r)r'r'r(r`Wsz Wigner9j.j34cCs |jdS)N)r&)r)r'r'r(ra[sz Wigner9j.j13cCs |jdS)N)r&)r)r'r'r(rb_sz Wigner9j.j24cCs |jdS)N)r&)r)r'r'r(r8csz Wigner9j.jcCstdd|jD S)NcSsg|] }|jqSr')r/)r0r1r'r'r(r2isz(Wigner9j.is_symbolic..)r3r&)r)r'r'r(r4gszWigner9j.is_symboliccs||j||j||jf||j||j||jf||j||j||j ffd}d}dgd}x0t dD]$t fddt dD|<qWd}xt dD]}d}xt dD]|} || } | d} | | } t | d| } t | d| } |dkr,| }qt |d|}t || }qW|dkr`|}qx t |D]} t |d}qjWt ||}qWt |jdd d }|S) Nr+r*r5r,csg|]}|qSr')r6)r0r7)r8r9r'r(r2wsz$Wigner9j._pretty..r:r]r^)r>r=)r;r r$rar"r_rbr[r`r8rr<r6rr=r>r?r@)r)rAr&rBrCrDrEr7rFrGrHrIrJrKr')r8r9r(rLlsB$ $   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\}) rr;r r"r[r$r_r`rarbr8rM)r)rAr&rNr'r'r(rOszWigner9j._latexc Ks8|jrtdt|j|j|j|j|j|j|j |j |j S)NzCoefficients must be numerical) r4rPrr r"r[r$r_r`rarbr8)r)rQr'r'r(rRsz Wigner9j.doitN)rSrTrUrVrrWr r"r[r$r_r`rarbr8r4rLrOrRr'r'r'r(r6s          &cCsft|trt|St|tr$t|St|trBtdd|jDSt|tr^tt|j |j S|SdS)aSimplify and combine CG coefficients 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')r)r0r1r'r'r(r2szcg_simp..N) isinstancer _cg_simp_addr _cg_simp_sumrr&r rbaseexp)er'r'r(rs    cCs g}g}t|}x|jD]}|trt|tr@|t|qt|trd}x.|jD]$}t|trr|t|9}qV||9}qVW|tr||q||q||q||qWt |\}}||t |\}}||t |\}}||t |t |S)aTakes a sum of terms involving Clebsch-Gordan coefficients and simplifies the terms. First, we create two lists, cg_part, which is all the terms involving CG coefficients, and other_part, which is all other terms. The cg_part list is then passed to the simplification methods, which return the new cg_part and any additional terms that are added to other_part r*) rr&Zhasrrfr appendrhr_check_varsh_871_1_check_varsh_871_2_check_varsh_872_9r)rkZcg_part other_partr1Ztermstermotherr'r'r(rgs2                  rgc Csttd\}}}}|t|||d||}d|dt|d}|t|}d|d}||} t|||||||||f||f|| S)N)aalphabltrr+r*)rrrrabs_check_cg_simp) term_listrsrtrurvexprsimpsign build_expr index_exprr'r'r(rms  rmc Csttd\}}}}|t|||| |d}td|dt|d}d|||t|}d|d}||} t|||||||||f||f|| S)N)rsrtcrvrr+r*r5)rrrr rrwrx) ryrsrtrrvrzr{r|r}r~r'r'r(rns rncCsttd\ }}}}}}}}} | t||||||d} d} | t| } t||} t||}||dt| | |kfdt| |f||| kf}|||}t| | | | |||||||| f||||f|| \}}t||} ||}|d| | |d}|| | |||}t| | | | |||||||| f||||f|| \}}t||||||t||||||} t||t||} td} t||} t||}||dt| | |kfdt| |f||| kf}|||}t| | | td|||||||||f||||||f|| \}}t||} ||}|d| | |d}|| | |||}t| | | td|||||||||f||||||f|| \}}||||fS)N) rsrtalphaprubetabetaprgammarvr+r*r) rrrrwr rrxrr)ryrsrtrrurrrrrvrzr{r|xyr}r~Zother1Zother2Zother3Zother4r'r'r(ro s:   2 2 2$  2 < <roc  sd} d} x| tkrt| |t|dkr@| d7} q t|jsZ| d7} q fdd|D} dg|} xt| tD]} t| || t|t|||| fd}|dkrqt|| |jsq| ||d| ||||| |f| || |<qWtdd| Drtd d| D}d d| D}|| fd d|DxB| D]:}t |d |kr |d ||d |dqW| |||7} q | d7} q W| 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')r0r)sub_1r'r(r2isz"_check_cg_simp..)r|css|]}|dk VqdS)Nr')r0r7r'r'r( rsz!_check_cg_simp..cSsg|]}t|dqS)r+)rw)r0rqr'r'r(r2sscSsg|] }|dqS)rr')r0rqr'r'r(r2tscsg|]}|qSr')pop)r0r8)ryr'r(r2wsr+r,) len _check_cgrsubsr/rr3minsortreverserwrl)rzr{r|rvryZ variablesZ dep_variablesZbuild_index_exprr~rpr7Zsub_depZcg_indexr8Zsub_2Zmin_ltindicesrqr')rryr(rx6s>)6F ( rxNcCs^||}|dkrdS|dk rJt|ts0td|d|d|ksJdSt||krZ|SdS)z2Checks whether a term matches the given expressionNzsign must be a tuplerr*)matchrfrM TypeErrorrr)Zcg_termrzlengthr|Zmatchesr'r'r(rs   rcCst|}t|}t|}|S)N)_check_varsh_sum_871_1_check_varsh_sum_871_2_check_varsh_sum_872_4)rkr'r'r(rhsrhc Csrtd}td}td}|tt|||d|||| |f}|dk rnt|dkrnd|dt|d|S|S)Nrsrtrurr+r*)rr rr rrrr)rkrsrtrurr'r'r(rs&rc Cstd}td}td}|td||t|||| |d|| |f}|dk rt|dkrtd|dt|d|S|S)Nrsrtrr5rr+r*) rr rr rrr rr)rkrsrtrrr'r'r(rs0 rc Cstd}td}td}td}td}td}td}td}|tt||||||t|||||||| |f|| |f} | dk rt| d krt||t||| S|tt||||||d || |f|| |f} | dk rt| d krd S|S) Nrsrtrurrcprgammaprer+rcr*)rrr rrrr) rkrsrtrurrrrrZmatch1Zmatch2r'r'r(rs"*.rcsttrfddfSgd}tts:tts:tdttrtjjrtjjrxfddtjDn fddfSttrx,j D]"}t|tr |q||9}qW||t |fSdS)Nr*z term must be CG, Add, Mul or Powcsg|]}jqSr')rlri)r0rK)cgrqr'r(r2sz_cg_list..) rfrrr NotImplementedErrorrrjr/rr&rlrw)rqZcoeffr1r')rrqr(_cg_lists          r)N).rVZ __future__rrZsympyrrrrrr r r r r rrZsympy.core.compatibilityrZ sympy.printing.pretty.stringpictrrZ(sympy.functions.special.tensor_functionsrZsympy.physics.wignerrrrr__all__rrrrrrgrmrnrorxrrhrrrrr'r'r'r(s68  xGYh*+  -K