Ed|0dZddlmZddlmZddlmZddlmZm Z m Z ddl m Z ddl mZddlmZdd lmZmZdd lmZdd lmZdd lmZmZdd lmZddlmZddlm Z dgZ!dZ"dZ#dZ$ddZ%ddZ&ddZ'ddZ(ddZ)GddeZ*dZ+dZ,dZ-dS) 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) 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 Copyright (C) 2008 Jens Rasch )Sum)Add)Function)IIntegerpi)S)Dummy)sympify)binomial factorial)exp)sqrt)cossin)Ynm)zeros)ImmutableMatrixc0|ttkr`tttt|dzD]-}tt|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 4lib/python3.11/site-packages/sympy/physics/wigner.py_calc_factlistrEs~6 S^^5IBF 44 5 5B   YrAv.3 4 4 4 4 \c"ggk\ ""c t t|dz|dzks2t|dz|dzkst|dz|dzkrtdt|dz|dzks2t|dz|dzkst|dz|dzkrtd||z|zdkr tjSt dt||z |z z}| }||z|z }|dkr tjS||z |z}|dkr tjS| |z|z} | dkr tjSt ||ks&t ||kst ||kr tjSt ||z|zdz|t |z|t |z|t |z} tt| t tt||z|z tt||z |zztt| |z|zztt||z ztt||zztt||z ztt||zztt||z ztt||zztt||z|zdzz } t| } | j s| j r| d} t | |z|z| |z|z d} t||z||z ||z|z }d}tt| t|dzD]}t|tt||z|z |z ztt||z|z ztt||z |z ztt||z|z |zztt||z|z |z z}|t d|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 `j_1 \ge |m_1|`, `j_2 \ge |m_2|`, `j_3 \ge |m_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 z(j values must be integer or half integerz(m values must be integer or half integerrr)r ValueErrorr Zerorabsmaxrrr is_complex is_infinite as_real_imagminr)j_1j_2j_3m_1m_2m_3prefida1a2a3maxfactargsqrtressqrtiminimaxsumresrdenress r wigner_3jr>fsl 37||sQwE#cAg,,#'"9E aLLC!G #ECDDD 37||sQwE#cAg,,#'"9E aLLC!G #ECDDD Sy3!v bSsS111 2 2F $C sSB Avv sSB Avv c B Avv  C3CHHsNC3v #)c/A%sSXX~sSXX~C.""G3w<<   iC#IO 4 45s39s?3345sC4#:#34456s39~~./s39~~. / s39~~. / s39~~. /s39~~./s39~~./00 #cCi#o)**+ ,G7mmG,W0,&&((+ tczC#c!11 5 5D sSy#)S3Y_ 5 5D FCIIs4yy1}--44m c"s(S.3.// 01 c#)b.)) *+ c#(S.)) *+ c"s(S.3.// 0 1 c#)c/B.// 0 1 '2"*--33 F V #C Jr c dt||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 r#r"r)r rr>)r,r-r.r/r0r1r=s rclebsch_gordanr@sUd '#)c/** *T!c'A+->-> >#sCcC400 1C Jr Nct||z|z ||z|z krtdt||z|z ||z|z krtdt||z|z ||z|z krtd||z|z dkr tjS||z|z dkr tjS||z|z dkr tjSt ||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-| | d}|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 r%r'rrrrevalfr*)aabbccprecr6r7r8s r_big_delta_coeffrG,s> 27R<R"Wr\*gefff 27R<R"Wr\*gefff 27R<R"Wr\*gefff R" v  R" v  R" v "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 rrr#) rGr r%r'r+rrrrr)rCrDrEddeeffrFprefacr9r:r6r;kkr<r=s rracahrNfsjb"b$ / /RT**+RT**+ RT**+F{v rBw|R"Wr\27R<b2 F FD rBw|b "r'B,"3R"Wr\B5F G GD$(BGbL2-rBw|b/@b2"$$G7 FCIIs4yy1}--HHBGbL2-../ c"r'B,+,, -. c"r'B,+,, -. c"r'B,+,, -. c"r'B,+b011 2 3 c"r'B,+b011 2 3 c"r'B,+b011 2 3'2"*ya/@"@AACGG 6/RCR" r(9$:$:: :C Jr c hdt||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]_. r#)rrN)r,r-r.j_4j_5j_6rFr=s r wigner_6jrSsEx #cCi#o+,, , c3S#sD11 2C Jr c Vtt||z||z||zdz} | dz} d} t| t| dzdD]U} | | dzt|||||| dz | zt|||||| dz | zt|||||| dz | zz} V| 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) # ==1/18 0.05555555... >>> wigner_9j(1/2,1/2,0, 1/2,3/2,1, 0,1,1, prec=64) # ==1/6 0.1666666... 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+rrN)r,r-r.rPrQrRj_7j_8j_9rFr:r9r;rMs r wigner_9jrXsf s39cCis33a7 8 8D !8D FD#d))a-++9926 #sCc264 8 89 #sCc264 8 89 #sCc264 8 899 Mr ct||ks&t||kst||krtdt||ks&t||kst||krtd||z|z}|dz}||z|z } | dkr tjS||z |z} | dkr tjS| |z|z} | dkr tjS|dzr tjS||z|zdkr tjSt ||ks&t ||kst ||kr tjSt | |z|z| |z|z d} t ||z||z ||z|z } t ||z|zdz| dz}t|d|zdzd|zdzzd|zdzzt||z zt||zzt||z zt||zzt||z zt||zzdtzz }t|}tt|t||z |zzt||z |zzt||z|z ztd|zdzz t||z t||z zt||z zz }d}tt| t| dzD]}t|t||z|z |z zt||z|z zt||z |z zt||z|z |zzt||z|z |z z}|td|z|z z}||z|ztd||z|z|z zz}|| |}|S)a* 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) 0.00689500421922113448... 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. zl values must be integerzm values must be integerr"rrr#)rr$r r%r&r'r+rrrrrrn)l_1l_2l_3r/r0r1rFsumLbigLr3r4r5r9r:r6r7r8rLr;rr<r=s rgauntraVsFP 3xx35#c((c/5SXX_53444 3xx35#c((c/5SXX_53444 9s?D 19D sSB Avv sSB Avv c B Avv  axv  c CAv  C3CHHsNC3v tczC#c!11 5 5D sSy#)S3Y_ 5 5D#)c/A%tax00G73w{q3w{+q3w{;#)(s346?c 6JK#)(s346?c 6JK 2G7mmG Yt_ysS'AAsSy3/02;C#IO2LMNN!d(Q,  4#:  4#:  !*4#:!6 78F FCIIs4yy1}--44miS3(<== cCi"n %&(1#(S.(AB b3hns* +,.7c C"8L.MN'2"*--33 F V #grtczC7G#7M.N&O&O OC eeDkk Jr ceZdZdZdS)Wigner3jc ^td|jDrt|jS|S)Nc3$K|] }|jV dSN) is_number).0objs r z Wigner3j.doit..s$22s}222222r )allargsr>)selfhintss rdoitz Wigner3j.doits5 22 222 2 2 di( (Kr N)__name__ __module__ __qualname__ror rrcrcs#r rcc t|}t|}t|}t|}t|}t|}td}d}tj||zzt t |||z||||||||zdz |dz|dzz |dzz |z|z |z z|t ||z ||zfzS)a) Returns dot product of rotational gradients of spherical harmonics. Explanation =========== This function returns the right hand side of the following expression: .. math :: \vec{R}Y{_j^{p}} \cdot \vec{R}Y{_l^{m}} = (-1)^{m+p} \sum\limits_{k=|l-j|}^{l+j}Y{_k^{m+p}} * \alpha_{l,m,j,p,k} * \frac{1}{2} (k^2-j^2-l^2+k-j-l) Arguments ========= j, p, l, m .... indices in spherical harmonics (expressions or integers) theta, phi .... angle arguments in spherical harmonics Example ======= >>> from sympy import symbols >>> from sympy.physics.wigner import dot_rot_grad_Ynm >>> theta, phi = symbols("theta phi") >>> dot_rot_grad_Ynm(3, 2, 2, 0, theta, phi).doit() 3*sqrt(55)*Ynm(5, 2, theta, phi)/(11*sqrt(pi)) kc td|zdzd|zdzzd|zdzzdtzz t|||tjtjtjzt|||||| |z zS)Nr"rrZ)rrrcr r%)lmjprus ralphazdot_rot_grad_Ynm..alpha$s|QqSUQqSUOQqSU+QrT233Aq!&!&!&99:Aq!Q1--. .r r")r r r NegativeOnerrr&)ryrzrwrxthetaphirur{s rdot_rot_grad_Ynmrs>  A A A A ENNE #,,C c A... MQqS !CAqsE3(?(?%%!AaPQBRBR(RUV(V Q$q!t)AqD. 1 Q ) "#S1XXqs!3%5%5 55r c V fdtdzdzD}tdzdz}t|D]\} t|D]\} t z z g}t d z g}t t zt z zt zz t z z } fdt||dzD} |t| z|||f<t|S)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 ⎦ cg|]}|z SrsrsrhiJs r z"wigner_d_small..###1###r r"rrc g|]}dz |z ztzz |z ztz |ztdz d|zzzzztdz dzd|zz z z zzS)r#r")r rr)rhsrMiMjbetas rrz"wigner_d_small..s;;;  AbDF^adAbDF++,adA&&'a[[1Q3r6"9-.a[[1Q3qs72:b=1 2;;;r ) rr enumerater'r+rr rr) rrMdrrysigmamaxsigmamindijtermsrrs `` @@rwigner_d_smallr-shd $###eAaCEll###A ac!e A1&&2q\\ & &EArRCFAbD>**HAqt9~~Hy2y26 2''02788C;;;;;;; $Hhqj99 ;;;E#u+oAadGG &" 1  r ct|fdtdzdzDfdtD}t|S)uReturn the Wigner D matrix for angular momentum J. Explanation =========== J : An integer, half-integer, or SymPy symbol for the total angular momentum of the angular momentum space being rotated. alpha, beta, gamma - Real numbers representing the Euler. Angles of rotation about the so-called vertical, line of nodes, and figure axes. See [Edmonds74]_. Returns ======= A matrix representing the corresponding Euler angle rotation( in the basis of eigenvectors of `J_z`). .. math :: \mathcal{D}_{\alpha \beta \gamma} = \exp\big( \frac{i\alpha}{\hbar} J_z\big) \exp\big( \frac{i\beta}{\hbar} J_y\big) \exp\big( \frac{i\gamma}{\hbar} J_z\big) The components are calculated using the general form [Edmonds74]_, equation 4.1.12. Examples ======== The simplest possible example: >>> from sympy.physics.wigner import wigner_d >>> from sympy import Integer, symbols, pprint >>> half = 1/Integer(2) >>> alpha, beta, gamma = symbols("alpha, beta, gamma", real=True) >>> pprint(wigner_d(half, alpha, beta, gamma), use_unicode=True) ⎡ ⅈ⋅α ⅈ⋅γ ⅈ⋅α -ⅈ⋅γ ⎤ ⎢ ─── ─── ─── ───── ⎥ ⎢ 2 2 ⎛β⎞ 2 2 ⎛β⎞ ⎥ ⎢ ℯ ⋅ℯ ⋅cos⎜─⎟ ℯ ⋅ℯ ⋅sin⎜─⎟ ⎥ ⎢ ⎝2⎠ ⎝2⎠ ⎥ ⎢ ⎥ ⎢ -ⅈ⋅α ⅈ⋅γ -ⅈ⋅α -ⅈ⋅γ ⎥ ⎢ ───── ─── ───── ───── ⎥ ⎢ 2 2 ⎛β⎞ 2 2 ⎛β⎞⎥ ⎢-ℯ ⋅ℯ ⋅sin⎜─⎟ ℯ ⋅ℯ ⋅cos⎜─⎟⎥ ⎣ ⎝2⎠ ⎝2⎠⎦ cg|]}|z Srsrsrs rrzwigner_d..rr r"rcXg|]$\fdtD%S)cg|]E\}}ttzz|fztt|zzzFSrs)rr)rhryrrr{rgammars rrz'wigner_d...s[ % % %a ad5j//!AqD' !#ad5j// 1 % % %r )r)rhrrrr{rrs @@rrzwigner_d..sp @ @ @).B % % % % % % % % || % % % @ @ @r )rrrr)rr{rrDrrs`` ` @@rwigner_drsf q$A####eAaCEll###A @ @ @ @ @ @ @2;A,, @ @ @A 1  r rf).__doc__sympy.concrete.summationsrsympy.core.addrsympy.core.functionrsympy.core.numbersrrrsympy.core.singletonr sympy.core.symbolr sympy.core.sympifyr (sympy.functions.combinatorial.factorialsr r &sympy.functions.elementary.exponentialr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrr+sympy.functions.special.spherical_harmonicsrsympy.matrices.densersympy.matrices.immutablerrrr>r@rGrNrSrXrarcrrrrsr rrs=..^*)))))((((((//////////""""""######&&&&&&JJJJJJJJ666666999999????????;;;;;;&&&&&&444444C ###BLLL^444n7777tNNNNb^^^^B;;;;|[[[[~x-5-5-5`FFFR77777r