B [[@sdZddlmZmZddlmZmZmZmZm Z m Z m Z m Z m Z ddlmZdgZddZdd Zd d Zdd dZdddZdddZdddZdddZGddde ZddZd S) 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 ~~~~~~~~~~ .. [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 )print_functiondivision) IntegerpisqrtsympifyDummySSumYnmFunction)rangecCsV|ttkrBx4tttt|dD]}tt|d|q$Wtdt|dS)a 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] rN)len _Factlistr intappend)Znniir3lib/python3.7/site-packages/sympy/physics/wigner.py_calc_factlist-s rcCsFt|d|dks>> 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 z(j values must be integer or half integerz(m values must be integer or half integerrr) r ValueErrorrabsmaxrrrZ is_complex as_real_imagminr )j_1j_2j_3m_1m_2m_3Zprefida1a2a3maxfactargsqrtressqrtiminimaxsumresrdenresrrr wigner_3jIsPX((   *$   ^ r/cCs<dt|||td|dt|||||| }|S)a Calculates the Clebsch-Gordan coefficient `\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \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:: \langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \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 rrr)rrr/)rrr r!r"r#r.rrrclebsch_gordans,"r0NcCsbt||||||kr$tdt||||||krHtdt||||||krltd|||dkrdS|||dkrdS|||dkrdSt||||||||||||d}t|ttt|||tt|||tt|||ttt|||d}t|}|r^||d}|S)ah 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) zJj values must be integer or half integer and fulfill the triangle relationrr) rrrrrrrZevalfr)aabbccprecr'r(r)rrr_big_delta_coeff s,2&r5cCst||||t||||t||||t||||}|dkrDdSt||||||||||||}t||||||||||||} t| d||||||||||||} t| d} xtt|t| dD]} tt| |||tt| |||tt| |||tt| |||tt||||| tt||||| tt||||| } | td| t| d| } qW|| dt||||}|S)a 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 rrr)r5rrrr rrr)r1r2r3ZddZeeZffr4prefacr*r+r'r,kkr-r.rrrracahAs 1(.0$$ r8c Cs0dt||||t|||||||}|S)a 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) r)rr8)rrr j_4j_5j_6r4r.rrr wigner_6js[r<c  Cstt||||||d} | d} d} xpt| t| ddD]X} | | dt|||||| d| t|||||| d| t|||||| d| } qBW| 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)`. 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]_. rrr)rrr r8)rrr r9r:r;Zj_7Zj_8Zj_9r4r+r*r,r7rrr wigner_9js/ 6 r=cCs"t||ks$t||ks$t||kr,tdt||ksPt||ksPt||krXtd|||}|d}|||} | dkrdS|||} | dkrdS| ||} | dkrdS|drdS|||dkrdSt||kst||kst||krdSt| ||| ||d} t|||||||} t|||d| d}t|d|dd|dd|dt||t||t||t||t||t||dt}t|}t t|t|||t|||t|||td|dt||t||t||}d}xt t| t| dD]~}t|t||||t|||t|||t||||t||||}|t d||}qdW|||t d||||}|dk r| |}|S) a 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 zl values must be integerzm values must be integerrrrrN) rrrrrrrrrrr n)Zl_1Zl_2Zl_3r!r"r#r4ZsumLZbigLr$r%r&r*r+r'r(r)r6r,rr-r.rrrgaunt&sLe$$   $ j J&d$  r@c@seZdZddZdS)Wigner3jcKs&tdd|jDrt|jS|SdS)Ncss|] }|jVqdS)N)Z is_number).0objrrr sz Wigner3j.doit..)allargsr/)selfZhintsrrrdoits z Wigner3j.doitN)__name__ __module__ __qualname__rHrrrrrAsrAc Cst|}t|}t|}t|}t|}t|}td}dd}td ||tt|||||||||||d|d|d|d||||t||||fS)a 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)) kc Ssftd|dd|dd|ddtt|||tdtdtdt|||||| |S)Nrrr>r)rrrAr )lmjprLrrralphaszdot_rot_grad_Ynm..alpharr)rrr r r r)rOrPrMrNZthetaZphirLrQrrrdot_rot_grad_YnmsrR)N)N)N)N)N)__doc__Z __future__rrZsympyrrrrrr r r r Zsympy.core.compatibilityr rrr/r0r5r8r<r=r@rArRrrrr!s , 1 6 K ` :