ó ¡¼™\c@ sïdZddlmZmZddlmZmZmZmZm Z m Z m Z m Z m Z ddlmZdgZd„Zd„Zd„Zdd „Zdd „Zdd „Zdd „Zdd „Zde fd„ƒYZd„ZdS(sŒ 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 ~~~~~~~~~~ .. [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) 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 Copyright (C) 2008 Jens Rasch iÿÿÿÿ(tprint_functiontdivision( tIntegertpitsqrttsympifytDummytStSumtYnmtFunction(trangeicC sj|ttƒkrXxCtttƒt|dƒƒD]}tjt|d|ƒq2Wntt|ƒd S(sþ Function calculates a list of precomputed factorials in order to massively accelerate future calculations of the various coefficients. INPUT: - ``nn`` - integer, highest factorial to be computed OUTPUT: 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] i(tlent _FactlistR tinttappend(tnntii((s3lib/python2.7/site-packages/sympy/physics/wigner.pyt_calc_factlist-s& cC søt|dƒ|dksNt|dƒ|dksNt|dƒ|dkr]tdƒ‚nt|dƒ|dks«t|dƒ|dks«t|dƒ|dkrºtdƒ‚n|||dkrÒdStdt|||ƒƒ}| }|||}|dkrdS|||}|dkr3dS| ||} | dkrRdSt|ƒ|ksˆt|ƒ|ksˆt|ƒ|krŒdSt|||d|t|ƒ|t|ƒ|t|ƒƒ} tt| ƒƒttt|||ƒtt|||ƒtt| ||ƒtt||ƒtt||ƒtt||ƒtt||ƒtt||ƒtt||ƒƒtt|||dƒ} t| ƒ} | jrÔ| j ƒd} nt| ||| ||dƒ} t |||||||ƒ}d}xÃt t| ƒt|ƒdƒD]¢}t|tt||||ƒtt|||ƒtt|||ƒtt||||ƒtt||||ƒ}|td|ƒ|}q@W| ||}|S(sn Calculate the Wigner 3j symbol `\operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)`. INPUT: - ``j_1``, ``j_2``, ``j_3``, ``m_1``, ``m_2``, ``m_3`` - integer or half integer OUTPUT: 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]_. REFERENCES: .. [Regge58] 'Symmetry Properties of Clebsch-Gordan Coefficients', T. Regge, Nuovo Cimento, Volume 10, pp. 544 (1958) .. [Edmonds74] 'Angular Momentum in Quantum Mechanics', A. R. Edmonds, Princeton University Press (1974) AUTHORS: - Jens Rasch (2009-03-24): initial version is(j values must be integer or half integers(m values must be integer or half integeriiÿÿÿÿi( Rt ValueErrorRtabstmaxRR Rt is_complext as_real_imagtminR (tj_1tj_2tj_3tm_1tm_2tm_3tprefidta1ta2ta3tmaxfacttargsqrttressqrttimintimaxtsumresRtdentres((s3lib/python2.7/site-packages/sympy/physics/wigner.pyt wigner_3jIsPX44   6,œ  $"&gcC sHdt|||ƒtd|dƒt|||||| ƒ}|S(s¤ Calculates the Clebsch-Gordan coefficient `\left\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \right\rangle`. The reference for this function is [Edmonds74]_. INPUT: - ``j_1``, ``j_2``, ``j_3``, ``m_1``, ``m_2``, ``m_3`` - integer or half integer OUTPUT: Rational number times the square root of a rational number. EXAMPLES:: >>> from sympy import S >>> from sympy.physics.wigner import clebsch_gordan >>> clebsch_gordan(S(3)/2, S(1)/2, 2, S(3)/2, S(1)/2, 2) 1 >>> clebsch_gordan(S(3)/2, S(1)/2, 1, S(3)/2, -S(1)/2, 1) sqrt(3)/2 >>> clebsch_gordan(S(3)/2, S(1)/2, 1, -S(1)/2, S(1)/2, 0) -sqrt(2)/2 NOTES: The Clebsch-Gordan coefficient will be evaluated via its relation to Wigner 3j symbols: .. math:: \left\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \right\rangle =(-1)^{j_1-j_2+m_3} \sqrt{2j_3+1} \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,-m_3) See also the documentation on Wigner 3j symbols which exhibit much higher symmetry relations than the Clebsch-Gordan coefficient. AUTHORS: - Jens Rasch (2009-03-24): initial version iÿÿÿÿii(RRR+(RRRRRRR*((s3lib/python2.7/site-packages/sympy/physics/wigner.pytclebsch_gordanÚs,'cC sºt|||ƒ|||kr1tdƒ‚nt|||ƒ|||krbtdƒ‚nt|||ƒ|||kr“tdƒ‚n|||dkr«dS|||dkrÃdS|||dkrÛdSt||||||||||||dƒ}t|ƒttt|||ƒtt|||ƒtt|||ƒƒttt|||dƒƒ}t|ƒ}|r¶|j|ƒjƒd}n|S(sh Calculates the Delta coefficient of the 3 angular momenta for Racah symbols. Also checks that the differences are of integer value. INPUT: - ``aa`` - first angular momentum, integer or half integer - ``bb`` - second angular momentum, integer or half integer - ``cc`` - third angular momentum, integer or half integer - ``prec`` - precision of the ``sqrt()`` calculation OUTPUT: double - Value of the Delta coefficient EXAMPLES:: sage: from sage.functions.wigner import _big_delta_coeff sage: _big_delta_coeff(1,1,1) 1/2*sqrt(1/6) sJj values must be integer or half integer and fulfill the triangle relationii( RRRRRR RtevalfR(taatbbtcctprecR#R$R%((s3lib/python2.7/site-packages/sympy/physics/wigner.pyt_big_delta_coeff s,"""9 +# cC sPt||||ƒt||||ƒt||||ƒt||||ƒ}|dkr^dSt||||||||||||ƒ}t||||||||||||ƒ} t| d||||||||||||ƒ} t| ƒd} xtt|ƒt| ƒdƒD]î} tt| |||ƒtt| |||ƒtt| |||ƒtt| |||ƒtt||||| ƒtt||||| ƒtt||||| ƒ} | td| t| dƒ| } q6W|| dt||||ƒ}|S(s” Calculate the Racah symbol `W(a,b,c,d;e,f)`. INPUT: - ``a``, ..., ``f`` - integer or half integer - ``prec`` - precision, default: ``None``. Providing a precision can drastically speed up the calculation. OUTPUT: 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 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 iiiÿÿÿÿ(R2RRRR RR R(R.R/R0tddteetffR1tprefacR&R'R#R(tkkR)R*((s3lib/python2.7/site-packages/sympy/physics/wigner.pytracahAs 18 56( &£!($c C s<dt||||ƒt|||||||ƒ}|S(s Calculate the Wigner 6j symbol `\operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6)`. INPUT: - ``j_1``, ..., ``j_6`` - integer or half integer - ``prec`` - precision, default: ``None``. Providing a precision can drastically speed up the calculation. OUTPUT: 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]_. REFERENCES: .. [Regge59] 'Symmetry Properties of Racah Coefficients', T. Regge, Nuovo Cimento, Volume 11, pp. 116 (1959) iÿÿÿÿ(RR8(RRRtj_4tj_5tj_6R1R*((s3lib/python2.7/site-packages/sympy/physics/wigner.pyt wigner_6jŒs[c  C sÑtt||||||ƒdƒ} | d} d} x’t| t| ƒddƒD]t} | | dt|||||| d| ƒt|||||| d| ƒt|||||| d| ƒ} qUW| S(s 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)`. INPUT: - ``j_1``, ..., ``j_9`` - integer or half integer - ``prec`` - precision, default: ``None``. Providing a precision can drastically speed up the calculation. OUTPUT: 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]_. iii(RRR R8(RRRR9R:R;tj_7tj_8tj_9R1R'R&R(R7((s3lib/python2.7/site-packages/sympy/physics/wigner.pyt wigner_9jìs/( #G(cC sºt|ƒ|ks6t|ƒ|ks6t|ƒ|krEtdƒ‚nt|ƒ|ks{t|ƒ|ks{t|ƒ|krŠtdƒ‚n|||}|d}|||} | dkrÀdS|||} | dkrÞdS| ||} | dkrýdS|dr dS|||dkr#dSt|ƒ|ksYt|ƒ|ksYt|ƒ|kr]dSt| ||| ||dƒ} t|||||||ƒ} t|||d| dƒ}t|ƒd|dd|dd|dt||t||t||t||t||t||dt}t|ƒ}t t|t|||t|||t|||ƒtd|dt||t||t||}d}x¥t t| ƒt| ƒdƒD]„}t|t||||t|||t|||t||||t||||}|t d|ƒ|}qèW|||t d||||ƒ}|dk r¶|j |ƒ}n|S( sœ Calculate the Gaunt coefficient. 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} INPUT: - ``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. OUTPUT: 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}` ALGORITHM: 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]_. REFERENCES: .. [Liberatodebrito82] 'FORTRAN program for the integral of three spherical harmonics', A. Liberato de Brito, Comput. Phys. Commun., Volume 25, pp. 81-85 (1982) AUTHORS: - Jens Rasch (2009-03-24): initial version for Sage sl values must be integersm values must be integeriiiiiÿÿÿÿN( RRRRRRR RRRR tNonetn(tl_1tl_2tl_3RRRR1tsumLtbigLR R!R"R&R'R#R$R%R6R(RR)R*((s3lib/python2.7/site-packages/sympy/physics/wigner.pytgaunt&sLe66     6$" k  M'&f( tWigner3jcB seZd„ZRS(cK s.td„|jDƒƒr&t|jŒS|SdS(Ncs s|]}|jVqdS(N(t is_number(t.0tobj((s3lib/python2.7/site-packages/sympy/physics/wigner.pys Ås(talltargsR+(tselfthints((s3lib/python2.7/site-packages/sympy/physics/wigner.pytdoitÄs (t__name__t __module__RQ(((s3lib/python2.7/site-packages/sympy/physics/wigner.pyRIÂsc C såt|ƒ}t|ƒ}t|ƒ}t|ƒ}t|ƒ}t|ƒ}tdƒ}d„}tdƒ ||tt|||||ƒ||||||ƒd|d|d|d||||t||ƒ||fƒS(s Returns dot product of rotational gradients of spherical harmonics. 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)) tkc S s{td|dd|dd|ddtƒt|||tdƒtdƒtdƒƒt|||||| |ƒS(Niiii(RRRIR(tltmtjtpRT((s3lib/python2.7/site-packages/sympy/physics/wigner.pytalphaîsii(RRRRR R(RWRXRURVtthetatphiRTRY((s3lib/python2.7/site-packages/sympy/physics/wigner.pytdot_rot_grad_YnmÊs        N(t__doc__t __future__RRtsympyRRRRRRRR R tsympy.core.compatibilityR R RR+R,RAR2R8R<R@RHRIR\(((s3lib/python2.7/site-packages/sympy/physics/wigner.pyt!s@   ‘ 1 6 K ` : œ