wgbdZddlmZddlmZddlmZddlmZddlm Z m Z m Z m Z m Z ddlmZddlmZdd lmZdd lmZmZdd lmZdd lmZdd lmZddlmZmZddl m!Z!ddl"m#Z#ddl$m%Z%ddl&m'Z'dgZ(dZ)dZ*dZ+dZ,d$dZ-d$dZ.d$dZ/d$dZ0d$dZ1d$dZ2Gdd eZ3d!Z4d"Z5d#Z6y)%a> Wigner, Clebsch-Gordan, Racah, and Gaunt coefficients Collection of functions for calculating Wigner 3j, 6j, 9j, Clebsch-Gordan, Racah as well as Gaunt coefficients exactly, all evaluating to a rational number times the square root of a rational number [Rasch03]_. Please see the description of the individual functions for further details and examples. References ========== .. [Regge58] 'Symmetry Properties of Clebsch-Gordan Coefficients', T. Regge, Nuovo Cimento, Volume 10, pp. 544 (1958) .. [Regge59] 'Symmetry Properties of Racah Coefficients', T. Regge, Nuovo Cimento, Volume 11, pp. 116 (1959) .. [Edmonds74] A. R. Edmonds. Angular momentum in quantum mechanics. Investigations in physics, 4.; Investigations in physics, no. 4. Princeton, N.J., Princeton University Press, 1957. .. [Rasch03] J. Rasch and A. C. H. Yu, 'Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients', SIAM J. Sci. Comput. Volume 25, Issue 4, pp. 1416-1428 (2003) .. [Liberatodebrito82] 'FORTRAN program for the integral of three spherical harmonics', A. Liberato de Brito, Comput. Phys. Commun., Volume 25, pp. 81-85 (1982) .. [Homeier96] 'Some Properties of the Coupling Coefficients of Real Spherical Harmonics and Their Relation to Gaunt Coefficients', H. H. H. Homeier and E. O. Steinborn J. Mol. Struct., Volume 368, pp. 31-37 (1996) Credits and Copyright ===================== This code was taken from Sage with the permission of all authors: https://groups.google.com/forum/#!topic/sage-devel/M4NZdu-7O38 Authors ======= - Jens Rasch (2009-03-24): initial version for Sage - Jens Rasch (2009-05-31): updated to sage-4.0 - Oscar Gerardo Lazo Arjona (2017-06-18): added Wigner D matrices - Phil Adam LeMaitre (2022-09-19): added real Gaunt coefficient Copyright (C) 2008 Jens Rasch )Sum)Add) int_valued)Function)FloatIIntegerpiRational)S)Dummy)sympify)binomial factorial)re)exp)sqrt)cossin)Ynm)zeros)ImmutableMatrix)as_intc|ttk\rLtttt|dzD]$}tj t|dz |z&tdt|dzS)a1 Function calculates a list of precomputed factorials in order to massively accelerate future calculations of the various coefficients. Parameters ========== nn : integer Highest factorial to be computed. Returns ======= list of integers : The list of precomputed factorials. Examples ======== Calculate list of factorials:: sage: from sage.functions.wigner import _calc_factlist sage: _calc_factlist(10) [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800] rN)len _Factlistrangeintappend)nniis Y/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/sympy/physics/wigner.py_calc_factlistr$Nsc6 S^IBF 4 5B   YrAv.3 4 5 \c"gk ""ct|tr|St|tur0|j r t|Sd|zj ry|St|t rC|j dk(r|j|j z S|j dk(r0|jSt|trtt|Std|z)z>return Python int unless value is half-int (then return float)rz)expecting integer or half-integer, got %s) isinstancertypefloat is_integerr qpr_int_or_halfint ValueError)values r#r.r.os% e     u:  eG   !L E8 $ 77a<77577? " WW\77N E5 !uU|,, @5H IIr%c Htt||||||g\}}}}}}||z|zdk7rtjS||z|z }|dkrtjS||z |z}|dkrtjS| |z|z}|dkrtjSt ||kDst ||kDst ||kDrtjSt ||z rt ||z rt ||z stjSt ||z|zdz|t |z|t |z|t |z} tt| ttt||z|z tt||z |zztt| |z|zztt||z ztt||zztt||z ztt||zztt||z ztt||zztt||z|zdzz } t| } | js | jr| jd} t | |z|z| |z|z d} t||z||z ||z|z } d}t!t| t| dzD]}t|tt||z|z |z ztt||z|z ztt||z |z ztt||z|z |zztt||z|z |z z}|td|z|z z}tdt||z |z z}| |z|z}|S)a Calculate the Wigner 3j symbol `\operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)`. Parameters ========== j_1, j_2, j_3, m_1, m_2, m_3 : Integer or half integer. Returns ======= Rational number times the square root of a rational number. Examples ======== >>> from sympy.physics.wigner import wigner_3j >>> wigner_3j(2, 6, 4, 0, 0, 0) sqrt(715)/143 >>> wigner_3j(2, 6, 4, 0, 0, 1) 0 It is an error to have arguments that are not integer or half integer values:: sage: wigner_3j(2.1, 6, 4, 0, 0, 0) Traceback (most recent call last): ... ValueError: j values must be integer or half integer sage: wigner_3j(2, 6, 4, 1, 0, -1.1) Traceback (most recent call last): ... ValueError: m values must be integer or half integer Notes ===== The Wigner 3j symbol obeys the following symmetry rules: - invariant under any permutation of the columns (with the exception of a sign change where `J:=j_1+j_2+j_3`): .. math:: \begin{aligned} \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3) &=\operatorname{Wigner3j}(j_3,j_1,j_2,m_3,m_1,m_2) \\ &=\operatorname{Wigner3j}(j_2,j_3,j_1,m_2,m_3,m_1) \\ &=(-1)^J \operatorname{Wigner3j}(j_3,j_2,j_1,m_3,m_2,m_1) \\ &=(-1)^J \operatorname{Wigner3j}(j_1,j_3,j_2,m_1,m_3,m_2) \\ &=(-1)^J \operatorname{Wigner3j}(j_2,j_1,j_3,m_2,m_1,m_3) \end{aligned} - invariant under space inflection, i.e. .. math:: \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3) =(-1)^J \operatorname{Wigner3j}(j_1,j_2,j_3,-m_1,-m_2,-m_3) - symmetric with respect to the 72 additional symmetries based on the work by [Regge58]_ - zero for `j_1`, `j_2`, `j_3` not fulfilling triangle relation - zero for `m_1 + m_2 + m_3 \neq 0` - zero for violating any one of the conditions `m_1 \in \{-|j_1|, \ldots, |j_1|\}`, `m_2 \in \{-|j_2|, \ldots, |j_2|\}`, `m_3 \in \{-|j_3|, \ldots, |j_3|\}` Algorithm ========= This function uses the algorithm of [Edmonds74]_ to calculate the value of the 3j symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. Authors ======= - Jens Rasch (2009-03-24): initial version rr)mapr.r Zeroabsrmaxr$rr rr is_complex is_infinite as_real_imagminr)j_1j_2j_3m_1m_2m_3a1a2a3maxfactargsqrtressqrtiminimaxsumresr"denprefidress r# wigner_3jrMsr$'(+S#sC'E$G Cc3S Sy3!vv sSB Avvv sSB Avvv c B Avvv  C3CHsNC3vv sSy ! sSy ! sSy !vv #)c/A%sSX~sSX~C."G3w< iC#IO 45s39s?345sC4#:#3456s39~./s39~. / s39~. / s39~. /s39~./s39~./0 #cCi#o)*+ ,G7mGW00&&(+ tczC#c!11 5D sSy#)S3Y_ 5D FCIs4y1}-4m c"s(S.3./ 01 c#)b.) *+ c#(S.) *+ c"s(S.3./ 0 1 c#)c/B./ 0 1 '2"*-334bSsS11 2F F V #C Jr%c vdt||z |zztd|zdzzt|||||| z}|S)a Calculates the Clebsch-Gordan coefficient. `\left\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \right\rangle`. The reference for this function is [Edmonds74]_. Parameters ========== j_1, j_2, j_3, m_1, m_2, m_3 : Integer or half integer. Returns ======= Rational number times the square root of a rational number. Examples ======== >>> from sympy import S >>> from sympy.physics.wigner import clebsch_gordan >>> clebsch_gordan(S(3)/2, S(1)/2, 2, S(3)/2, S(1)/2, 2) 1 >>> clebsch_gordan(S(3)/2, S(1)/2, 1, S(3)/2, -S(1)/2, 1) sqrt(3)/2 >>> clebsch_gordan(S(3)/2, S(1)/2, 1, -S(1)/2, S(1)/2, 0) -sqrt(2)/2 Notes ===== The Clebsch-Gordan coefficient will be evaluated via its relation to Wigner 3j symbols: .. math:: \left\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \right\rangle =(-1)^{j_1-j_2+m_3} \sqrt{2j_3+1} \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,-m_3) See also the documentation on Wigner 3j symbols which exhibit much higher symmetry relations than the Clebsch-Gordan coefficient. Authors ======= - Jens Rasch (2009-03-24): initial version r2r'r)rrrM)r;r<r=r>r?r@rLs r#clebsch_gordanrOsLd '#)c/* *T!c'A+-> >#sCcC40 1C Jr%Nct||z|z s tdt||z|z s tdt||z|z s td||z|z dkrtjS||z|z dkrtjS||z|z dkrtjSt ||z|z ||z|z ||z|z ||z|zdz}t |t tt||z|z tt||z|z ztt||z|z zt tt||z|zdzz }t|}|r"|j|jd}|S)a Calculates the Delta coefficient of the 3 angular momenta for Racah symbols. Also checks that the differences are of integer value. Parameters ========== aa : First angular momentum, integer or half integer. bb : Second angular momentum, integer or half integer. cc : Third angular momentum, integer or half integer. prec : Precision of the ``sqrt()`` calculation. Returns ======= double : Value of the Delta coefficient. Examples ======== sage: from sage.functions.wigner import _big_delta_coeff sage: _big_delta_coeff(1,1,1) 1/2*sqrt(1/6) zJj values must be integer or half integer and fulfill the triangle relationrr) rr/r r4r6r$r rrrevalfr9)aabbccprecrDrErFs r#_big_delta_coeffrVKsB b2gl #eff b2gl #eff b2gl #eff R" vv  R" vv  R" vv "r'B,R" b2glBGbL1>> from sympy.physics.wigner import racah >>> racah(3,3,3,3,3,3) -1/14 Notes ===== The Racah symbol is related to the Wigner 6j symbol: .. math:: \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6) Please see the 6j symbol for its much richer symmetries and for additional properties. Algorithm ========= This function uses the algorithm of [Edmonds74]_ to calculate the value of the 6j symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. Authors ======= - Jens Rasch (2009-03-24): initial version rrr2) rVr r4r6r:r$rrrr )rRrSrTddeeffrUprefacrGrHrDrIkkrJrLs r#racahr]swjb"b$ /RT*+RT*+ RT*+F{vv rBw|R"Wr\27R<b2 FD rBw|b "r'B,"3R"Wr\B5F GD$(BGbL2-rBw|b/@b2"$G7 FCIs4y1}-HBGbL2-./ c"r'B,+, -. c"r'B,+, -. c"r'B,+, -. c"r'B,+b01 2 3 c"r'B,+b01 2 3 c"r'B,+b01 2 3'2"*ya/@"@ACGGH 6/RCR" r(9$:: :C Jr%c Xdt||z|z|zzt|||||||z}|S)a+ Calculate the Wigner 6j symbol `\operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6)`. Parameters ========== j_1, ..., j_6 : Integer or half integer. prec : Precision, default: ``None``. Providing a precision can drastically speed up the calculation. Returns ======= Rational number times the square root of a rational number (if ``prec=None``), or real number if a precision is given. Examples ======== >>> from sympy.physics.wigner import wigner_6j >>> wigner_6j(3,3,3,3,3,3) -1/14 >>> wigner_6j(5,5,5,5,5,5) 1/52 It is an error to have arguments that are not integer or half integer values or do not fulfill the triangle relation:: sage: wigner_6j(2.5,2.5,2.5,2.5,2.5,2.5) Traceback (most recent call last): ... ValueError: j values must be integer or half integer and fulfill the triangle relation sage: wigner_6j(0.5,0.5,1.1,0.5,0.5,1.1) Traceback (most recent call last): ... ValueError: j values must be integer or half integer and fulfill the triangle relation Notes ===== The Wigner 6j symbol is related to the Racah symbol but exhibits more symmetries as detailed below. .. math:: \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6) The Wigner 6j symbol obeys the following symmetry rules: - Wigner 6j symbols are left invariant under any permutation of the columns: .. math:: \begin{aligned} \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) &=\operatorname{Wigner6j}(j_3,j_1,j_2,j_6,j_4,j_5) \\ &=\operatorname{Wigner6j}(j_2,j_3,j_1,j_5,j_6,j_4) \\ &=\operatorname{Wigner6j}(j_3,j_2,j_1,j_6,j_5,j_4) \\ &=\operatorname{Wigner6j}(j_1,j_3,j_2,j_4,j_6,j_5) \\ &=\operatorname{Wigner6j}(j_2,j_1,j_3,j_5,j_4,j_6) \end{aligned} - They are invariant under the exchange of the upper and lower arguments in each of any two columns, i.e. .. math:: \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) =\operatorname{Wigner6j}(j_1,j_5,j_6,j_4,j_2,j_3) =\operatorname{Wigner6j}(j_4,j_2,j_6,j_1,j_5,j_3) =\operatorname{Wigner6j}(j_4,j_5,j_3,j_1,j_2,j_6) - additional 6 symmetries [Regge59]_ giving rise to 144 symmetries in total - only non-zero if any triple of `j`'s fulfill a triangle relation Algorithm ========= This function uses the algorithm of [Edmonds74]_ to calculate the value of the 6j symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. r2)rr])r;r<r=j_4j_5j_6rUrLs r# wigner_6jrbs?x #cCi#o+, , c3S#sD1 2C Jr%c "tt||z||z||zdz} | dz} d} t| t| dzdD]I} | | dzt|||||| dz | zt|||||| dz | zt|||||| dz | zz} K| S)a Calculate the Wigner 9j symbol `\operatorname{Wigner9j}(j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8,j_9)`. Parameters ========== j_1, ..., j_9 : Integer or half integer. prec : precision, default ``None``. Providing a precision can drastically speed up the calculation. Returns ======= Rational number times the square root of a rational number (if ``prec=None``), or real number if a precision is given. Examples ======== >>> from sympy.physics.wigner import wigner_9j >>> wigner_9j(1,1,1, 1,1,1, 1,1,0, prec=64) 0.05555555555555555555555555555555555555555555555555555555555555555 >>> wigner_9j(1/2,1/2,0, 1/2,3/2,1, 0,1,1, prec=64) 0.1666666666666666666666666666666666666666666666666666666666666667 It is an error to have arguments that are not integer or half integer values or do not fulfill the triangle relation:: sage: wigner_9j(0.5,0.5,0.5, 0.5,0.5,0.5, 0.5,0.5,0.5,prec=64) Traceback (most recent call last): ... ValueError: j values must be integer or half integer and fulfill the triangle relation sage: wigner_9j(1,1,1, 0.5,1,1.5, 0.5,1,2.5,prec=64) Traceback (most recent call last): ... ValueError: j values must be integer or half integer and fulfill the triangle relation Algorithm ========= This function uses the algorithm of [Edmonds74]_ to calculate the value of the 3j symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. r'rr)rr:rr])r;r<r=r_r`raj_7j_8j_9rUrHrGrIr\s r# wigner_9jrg9sf s39cCis3a7 8D !8D FD#d)a-+926 #sCc264 89 #sCc264 89 #sCc264 8999 Mr%c2||||||fDcgc] }t|c}\}}}}}}||z|z dkrtjS||z |zdkrtjS| |z|zdkrtjS||z|zdk7rtjSt||kDst||kDst||kDrtjSt ||z|zd\}} | dzrtjSt | |z|z| |z|z d} t ||z||z ||z|z } tt ||z|zdz| dztd|zdzd|zdzzd|zdzzt||z zt||zzt||z zt||zzt||z zt||zzdtzz } tt|t||z |zzt||z |zzt||z|z ztd|zdzz t||z t||z zt||z zz } d}tt| t| dzD]x}t|t||z|z |z zt||z|z zt||z |z zt||z|z |zzt||z|z |z z}|td|z|z z}z| | z|ztd||z|z|z zz}||j|}|Scc}w)aU Calculate the Gaunt coefficient. Explanation =========== The Gaunt coefficient is defined as the integral over three spherical harmonics: .. math:: \begin{aligned} \operatorname{Gaunt}(l_1,l_2,l_3,m_1,m_2,m_3) &=\int Y_{l_1,m_1}(\Omega) Y_{l_2,m_2}(\Omega) Y_{l_3,m_3}(\Omega) \,d\Omega \\ &=\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \operatorname{Wigner3j}(l_1,l_2,l_3,0,0,0) \operatorname{Wigner3j}(l_1,l_2,l_3,m_1,m_2,m_3) \end{aligned} Parameters ========== l_1, l_2, l_3, m_1, m_2, m_3 : Integer. prec - precision, default: ``None``. Providing a precision can drastically speed up the calculation. Returns ======= Rational number times the square root of a rational number (if ``prec=None``), or real number if a precision is given. Examples ======== >>> from sympy.physics.wigner import gaunt >>> gaunt(1,0,1,1,0,-1) -1/(2*sqrt(pi)) >>> gaunt(1000,1000,1200,9,3,-12).n(64) 0.006895004219221134484332976156744208248842039317638217822322799675 It is an error to use non-integer values for `l` and `m`:: sage: gaunt(1.2,0,1.2,0,0,0) Traceback (most recent call last): ... ValueError: l values must be integer sage: gaunt(1,0,1,1.1,0,-1.1) Traceback (most recent call last): ... ValueError: m values must be integer Notes ===== The Gaunt coefficient obeys the following symmetry rules: - invariant under any permutation of the columns .. math:: \begin{aligned} Y(l_1,l_2,l_3,m_1,m_2,m_3) &=Y(l_3,l_1,l_2,m_3,m_1,m_2) \\ &=Y(l_2,l_3,l_1,m_2,m_3,m_1) \\ &=Y(l_3,l_2,l_1,m_3,m_2,m_1) \\ &=Y(l_1,l_3,l_2,m_1,m_3,m_2) \\ &=Y(l_2,l_1,l_3,m_2,m_1,m_3) \end{aligned} - invariant under space inflection, i.e. .. math:: Y(l_1,l_2,l_3,m_1,m_2,m_3) =Y(l_1,l_2,l_3,-m_1,-m_2,-m_3) - symmetric with respect to the 72 Regge symmetries as inherited for the `3j` symbols [Regge58]_ - zero for `l_1`, `l_2`, `l_3` not fulfilling triangle relation - zero for violating any one of the conditions: `l_1 \ge |m_1|`, `l_2 \ge |m_2|`, `l_3 \ge |m_3|` - non-zero only for an even sum of the `l_i`, i.e. `L = l_1 + l_2 + l_3 = 2n` for `n` in `\mathbb{N}` Algorithms ========== This function uses the algorithm of [Liberatodebrito82]_ to calculate the value of the Gaunt coefficient exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. Authors ======= Jens Rasch (2009-03-24): initial version for Sage. rr'rr2)rr r4r5divmodr6r:r$rrr r rrn)l_1l_2l_3r>r?r@rUibigLremLrGrHrFr[rIr"rJrLs r#gauntrrwsR c3S9$;q $; Cc3S Sy3vv  Sy3vv  tczC!vv  c CAvv  C3CHsNC3vv c C+JD$ axvv tczC#c!11 5D sSy#)S3Y_ 5D3sSy3*D1H56AGaKAGaK0AGaK@#)(s346?c 6JK#)(s346?c 6JK 2G Yt_ysS'AAsSy3/02;C#IO2LMN!d(Q,  4#:  4#:  !*4#:!6 78F FCIs4y1}-4miS3(<== cCi"n %&(1#(S.(AB b3hns* +,.7c C"8L.MN'2"*-33 4 F V #grtczC7G#7M.N&O OC eeDk JW$;sLc ||||||fDcgc] }t|c}\}}}}}}td|||fDdzrtjS||z|zdzrtjS||z}t t ||z t t ||zt ||z } | |zdzr| dz } | t|| dz dvrtjSddfdfdfd } d } t| |dzD]_}| ||} t| |dzD]A} | | |}| | | z |}| t| |z|zt||||| | | z zz} Ca|| j|} | Scc}w) aX Calculate the real Gaunt coefficient. Explanation =========== The real Gaunt coefficient is defined as the integral over three real spherical harmonics: .. math:: \begin{aligned} \operatorname{RealGaunt}(l_1,l_2,l_3,m_1,m_2,m_3) &=\int Z^{m_1}_{l_1}(\Omega) Z^{m_2}_{l_2}(\Omega) Z^{m_3}_{l_3}(\Omega) \,d\Omega \\ \end{aligned} Alternatively, it can be defined in terms of the standard Gaunt coefficient by relating the real spherical harmonics to the standard spherical harmonics via a unitary transformation `U`, i.e. `Z^{m}_{l}(\Omega)=\sum_{m'}U^{m}_{m'}Y^{m'}_{l}(\Omega)` [Homeier96]_. The real Gaunt coefficient is then defined as .. math:: \begin{aligned} \operatorname{RealGaunt}(l_1,l_2,l_3,m_1,m_2,m_3) &=\int Z^{m_1}_{l_1}(\Omega) Z^{m_2}_{l_2}(\Omega) Z^{m_3}_{l_3}(\Omega) \,d\Omega \\ &=\sum_{m'_1 m'_2 m'_3} U^{m_1}_{m'_1}U^{m_2}_{m'_2}U^{m_3}_{m'_3} \operatorname{Gaunt}(l_1,l_2,l_3,m'_1,m'_2,m'_3) \end{aligned} The unitary matrix `U` has components .. math:: \begin{aligned} U^m_{m'} = \delta_{|m||m'|}*(\delta_{m'0}\delta_{m0} + \frac{1}{\sqrt{2}}\big[\Theta(m) \big(\delta_{m'm}+(-1)^{m'}\delta_{m'-m}\big)+i\Theta(-m)\big((-1)^{-m} \delta_{m'-m}-\delta_{m'm}*(-1)^{m'-m}\big)\big]) \end{aligned} where `\delta_{ij}` is the Kronecker delta symbol and `\Theta` is a step function defined as .. math:: \begin{aligned} \Theta(x) = \begin{cases} 1 \,\text{for}\, x > 0 \\ 0 \,\text{for}\, x \leq 0 \end{cases} \end{aligned} Parameters ========== l_1, l_2, l_3, m_1, m_2, m_3 : Integer. prec - precision, default: ``None``. Providing a precision can drastically speed up the calculation. Returns ======= Rational number times the square root of a rational number. Examples ======== >>> from sympy.physics.wigner import real_gaunt >>> real_gaunt(2,2,4,-1,-1,0) -2/(7*sqrt(pi)) >>> real_gaunt(10,10,20,-9,-9,0).n(64) -0.00002480019791932209313156167176797577821140084216297395518482071448 It is an error to use non-integer values for `l` and `m`:: real_gaunt(2.8,0.5,1.3,0,0,0) Traceback (most recent call last): ... ValueError: l values must be integer real_gaunt(2,2,4,0.7,1,-3.4) Traceback (most recent call last): ... ValueError: m values must be integer Notes ===== The real Gaunt coefficient inherits from the standard Gaunt coefficient, the invariance under any permutation of the pairs `(l_i, m_i)` and the requirement that the sum of the `l_i` be even to yield a non-zero value. It also obeys the following symmetry rules: - zero for `l_1`, `l_2`, `l_3` not fulfiling the condition `l_1 \in \{l_{\text{max}}, l_{\text{max}}-2, \ldots, l_{\text{min}}\}`, where `l_{\text{max}} = l_2+l_3`, .. math:: \begin{aligned} l_{\text{min}} = \begin{cases} \kappa(l_2, l_3, m_2, m_3) & \text{if}\, \kappa(l_2, l_3, m_2, m_3) + l_{\text{max}}\, \text{is even} \\ \kappa(l_2, l_3, m_2, m_3)+1 & \text{if}\, \kappa(l_2, l_3, m_2, m_3) + l_{\text{max}}\, \text{is odd}\end{cases} \end{aligned} and `\kappa(l_2, l_3, m_2, m_3) = \max{\big(|l_2-l_3|, \min{\big(|m_2+m_3|, |m_2-m_3|\big)}\big)}` - zero for an odd number of negative `m_i` Algorithms ========== This function uses the algorithms of [Homeier96]_ and [Rasch03]_ to calculate the value of the real Gaunt coefficient exactly. Note that the formula used in [Rasch03]_ contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [Rasch03]_. However, this function can in principle use any algorithm that computes the Gaunt coefficient, so it is suitable for finite precision arithmetic in so far as the algorithm which computes the Gaunt coefficient is. c3,K|] }|dks dyw)rrN).0ros r# zreal_gaunt..s /Q1 /s r'rc||k(rdSdS)Nrrru)rojs r#zreal_gaunt..saAQr%c|dzrdSdS)Nr'r2rru)es r#r{zreal_gaunt..sA"1r%cf|dkr*|| ||z z|| |zzSdSNrruabkron_delss r#r{zreal_gaunt..sI 1u 1~oa!f,xA2d0*+r%cL|dkDr|||| |zzSdSrrurs r#r{zreal_gaunt..s-Ahq!nxA2qt';;1r%ct|t||d|dz||t||zztdz zzS)Nrr')r5rr)rrABrs r#r{zreal_gaunt..sVXc!fc!f-x1~hq!n/L+,Q7QqAwY+>Q*G0HIr%r) rsumr r4r6r5r:rrrrrk)rlrmrnr>r?r@rUrolmaxlminCugntU1rzU2U3rrrrs @@@@r# real_gauntr sr c3S9$;q $; Cc3S /sCo //!3vv  c C1vv 9D s39~s3sSy>3sSy>B CD t q   5tax,,vv .H$A ,ALA IA D C4Q H q#YtSU# HA1cBA2a4B"RU2X,uS#sAq1"Q$'GGGD HH vvd| KA$;sE>ceZdZdZy)Wigner3jc btd|jDrt|jS|S)Nc34K|]}|jywN) is_number)rvobjs r#rwz Wigner3j.doit..s2s}}2s)allargsrM)selfhintss r#doitz Wigner3j.doits( 2 2 2dii( (Kr%N)__name__ __module__ __qualname__rrur%r#rrsr%rc t|}t|}t|}t|}t|}t|}td}d}tj||zzt t |||z||||||||zdz |dz|dzz |dzz |z|z |z z|t ||z ||zfzS)a) Returns dot product of rotational gradients of spherical harmonics. Explanation =========== This function returns the right hand side of the following expression: .. math :: \vec{R}Y{_j^{p}} \cdot \vec{R}Y{_l^{m}} = (-1)^{m+p} \sum\limits_{k=|l-j|}^{l+j}Y{_k^{m+p}} * \alpha_{l,m,j,p,k} * \frac{1}{2} (k^2-j^2-l^2+k-j-l) Arguments ========= j, p, l, m .... indices in spherical harmonics (expressions or integers) theta, phi .... angle arguments in spherical harmonics Example ======= >>> from sympy import symbols >>> from sympy.physics.wigner import dot_rot_grad_Ynm >>> theta, phi = symbols("theta phi") >>> dot_rot_grad_Ynm(3, 2, 2, 0, theta, phi).doit() 3*sqrt(55)*Ynm(5, 2, theta, phi)/(11*sqrt(pi)) kc td|zdzd|zdzzd|zdzzdtzz t|||tjtjtjzt|||||| |z zS)Nr'rri)rr rr r4)lmrzr-rs r#alphazdot_rot_grad_Ynm..alphasyQqSUQqSUOQqSU+QrT23Aq!&&!&&!&&9:Aq!Q1-. .r%r')rr r NegativeOnerrr5)rzr-rrthetaphirrs r#dot_rot_grad_Ynmrs>  A A A A ENE #,C c A. MMQqS !CAqsE3(?%!AaPQBR(RUV(V Q$q!t)AqD. 1 Q ) "#S1Xqs!3%5 55r%c td|zdzDcgc]}||z  }}td|zdz}t|D]\}}t|D]\}}t||z ||z g}t d| |z g} t t ||zt ||z zt ||zz t ||z z } t| |dzD cgc]o} d||z | z zt||z||z | z zt||z | zt|dz d| z|z|zzzt|dz d|zd| zz |z |z zzq} } | t| z|||f<t|Scc}wcc} w)uReturn the small Wigner d matrix for angular momentum J. Explanation =========== J : An integer, half-integer, or SymPy symbol for the total angular momentum of the angular momentum space being rotated. beta : A real number representing the Euler angle of rotation about the so-called line of nodes. See [Edmonds74]_. Returns ======= A matrix representing the corresponding Euler angle rotation( in the basis of eigenvectors of `J_z`). .. math :: \mathcal{d}_{\beta} = \exp\big( \frac{i\beta}{\hbar} J_y\big) The components are calculated using the general form [Edmonds74]_, equation 4.1.15. Examples ======== >>> from sympy import Integer, symbols, pi, pprint >>> from sympy.physics.wigner import wigner_d_small >>> half = 1/Integer(2) >>> beta = symbols("beta", real=True) >>> pprint(wigner_d_small(half, beta), use_unicode=True) ⎡ ⎛β⎞ ⎛β⎞⎤ ⎢cos⎜─⎟ sin⎜─⎟⎥ ⎢ ⎝2⎠ ⎝2⎠⎥ ⎢ ⎥ ⎢ ⎛β⎞ ⎛β⎞⎥ ⎢-sin⎜─⎟ cos⎜─⎟⎥ ⎣ ⎝2⎠ ⎝2⎠⎦ >>> pprint(wigner_d_small(2*half, beta), use_unicode=True) ⎡ 2⎛β⎞ ⎛β⎞ ⎛β⎞ 2⎛β⎞ ⎤ ⎢ cos ⎜─⎟ √2⋅sin⎜─⎟⋅cos⎜─⎟ sin ⎜─⎟ ⎥ ⎢ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎥ ⎢ ⎥ ⎢ ⎛β⎞ ⎛β⎞ 2⎛β⎞ 2⎛β⎞ ⎛β⎞ ⎛β⎞⎥ ⎢-√2⋅sin⎜─⎟⋅cos⎜─⎟ - sin ⎜─⎟ + cos ⎜─⎟ √2⋅sin⎜─⎟⋅cos⎜─⎟⎥ ⎢ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠⎥ ⎢ ⎥ ⎢ 2⎛β⎞ ⎛β⎞ ⎛β⎞ 2⎛β⎞ ⎥ ⎢ sin ⎜─⎟ -√2⋅sin⎜─⎟⋅cos⎜─⎟ cos ⎜─⎟ ⎥ ⎣ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎦ From table 4 in [Edmonds74]_ >>> pprint(wigner_d_small(half, beta).subs({beta:pi/2}), use_unicode=True) ⎡ √2 √2⎤ ⎢ ── ──⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢-√2 √2⎥ ⎢──── ──⎥ ⎣ 2 2 ⎦ >>> pprint(wigner_d_small(2*half, beta).subs({beta:pi/2}), ... use_unicode=True) ⎡ √2 ⎤ ⎢1/2 ── 1/2⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢-√2 √2 ⎥ ⎢──── 0 ── ⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢ -√2 ⎥ ⎢1/2 ──── 1/2⎥ ⎣ 2 ⎦ >>> pprint(wigner_d_small(3*half, beta).subs({beta:pi/2}), ... use_unicode=True) ⎡ √2 √6 √6 √2⎤ ⎢ ── ── ── ──⎥ ⎢ 4 4 4 4 ⎥ ⎢ ⎥ ⎢-√6 -√2 √2 √6⎥ ⎢──── ──── ── ──⎥ ⎢ 4 4 4 4 ⎥ ⎢ ⎥ ⎢ √6 -√2 -√2 √6⎥ ⎢ ── ──── ──── ──⎥ ⎢ 4 4 4 4 ⎥ ⎢ ⎥ ⎢-√2 √6 -√6 √2⎥ ⎢──── ── ──── ──⎥ ⎣ 4 4 4 4 ⎦ >>> pprint(wigner_d_small(4*half, beta).subs({beta:pi/2}), ... use_unicode=True) ⎡ √6 ⎤ ⎢1/4 1/2 ── 1/2 1/4⎥ ⎢ 4 ⎥ ⎢ ⎥ ⎢-1/2 -1/2 0 1/2 1/2⎥ ⎢ ⎥ ⎢ √6 √6 ⎥ ⎢ ── 0 -1/2 0 ── ⎥ ⎢ 4 4 ⎥ ⎢ ⎥ ⎢-1/2 1/2 0 -1/2 1/2⎥ ⎢ ⎥ ⎢ √6 ⎥ ⎢1/4 -1/2 ── -1/2 1/4⎥ ⎣ 4 ⎦ r'rrr2) rr enumerater:r6rrrrrrr) JbetaroMdMirzMjsigmamaxsigmamindijrtermss r#wigner_d_smallrsdAaCEl#1#A# ac!e A1&2q\ &EArAbD!B$<(HAs2v;'Hy2y26 2''0278C$Hhqj9 ;  AbDF^adAbDF+,adA&'a[1Q3r6"9-.a[1Q3qs72:b=1 2;E;#u+oAadG &&$ 1 ) $;s E'A4E,ctt||}td|zdzDcgc]}||z  }}t|D cgc]T\}}t|D cgc]8\}} tt|z|z|||fztt| z|zz:c} }V} }}}} t | Scc}wcc} }wcc} }}}w)uReturn the Wigner D matrix for angular momentum J. Explanation =========== J : An integer, half-integer, or SymPy symbol for the total angular momentum of the angular momentum space being rotated. alpha, beta, gamma - Real numbers representing the Euler. Angles of rotation about the so-called vertical, line of nodes, and figure axes. See [Edmonds74]_. Returns ======= A matrix representing the corresponding Euler angle rotation( in the basis of eigenvectors of `J_z`). .. math :: \mathcal{D}_{\alpha \beta \gamma} = \exp\big( \frac{i\alpha}{\hbar} J_z\big) \exp\big( \frac{i\beta}{\hbar} J_y\big) \exp\big( \frac{i\gamma}{\hbar} J_z\big) The components are calculated using the general form [Edmonds74]_, equation 4.1.12. Examples ======== The simplest possible example: >>> from sympy.physics.wigner import wigner_d >>> from sympy import Integer, symbols, pprint >>> half = 1/Integer(2) >>> alpha, beta, gamma = symbols("alpha, beta, gamma", real=True) >>> pprint(wigner_d(half, alpha, beta, gamma), use_unicode=True) ⎡ ⅈ⋅α ⅈ⋅γ ⅈ⋅α -ⅈ⋅γ ⎤ ⎢ ─── ─── ─── ───── ⎥ ⎢ 2 2 ⎛β⎞ 2 2 ⎛β⎞ ⎥ ⎢ ℯ ⋅ℯ ⋅cos⎜─⎟ ℯ ⋅ℯ ⋅sin⎜─⎟ ⎥ ⎢ ⎝2⎠ ⎝2⎠ ⎥ ⎢ ⎥ ⎢ -ⅈ⋅α ⅈ⋅γ -ⅈ⋅α -ⅈ⋅γ ⎥ ⎢ ───── ─── ───── ───── ⎥ ⎢ 2 2 ⎛β⎞ 2 2 ⎛β⎞⎥ ⎢-ℯ ⋅ℯ ⋅sin⎜─⎟ ℯ ⋅ℯ ⋅cos⎜─⎟⎥ ⎣ ⎝2⎠ ⎝2⎠⎦ r'r)rrrrrr) rrrgammarrorrrzrDs r#wigner_drisf q$AAaCEl#1#A#2;A, @ @).By| %a ad5j/!AqD' !#ad5j/ 1 % @A @ 1  $ % @s B'B2 =B,B2 ,B2 r)7__doc__sympy.concrete.summationsrsympy.core.addrsympy.core.numbersrsympy.core.functionrrrr r r sympy.core.singletonr sympy.core.symbolr sympy.core.sympifyr(sympy.functions.combinatorial.factorialsrr$sympy.functions.elementary.complexesr&sympy.functions.elementary.exponentialr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrr+sympy.functions.special.spherical_harmonicsrsympy.matrices.densersympy.matrices.immutablersympy.utilities.miscrrr$r.rMrOrVr]rbrgrrrrrrrrur%r#rs4j*)(@@"#&J369?;&4'C #BJ&Od4n9xNb^B;|SlXvx-5`FR7r%