# Source code for sympy.physics.wigner

r"""
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)

~~~~~~~~~~~~~~~~~~~~~

This code was taken from Sage with the permission of all authors:

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 <jyr2000@gmail.com>
"""
from __future__ import print_function, division

from sympy import Integer, pi, sqrt, sympify
#from sage.rings.complex_number import ComplexNumber
#from sage.rings.finite_rings.integer_mod import Mod

# This list of precomputed factorials is needed to massively
# accelerate future calculations of the various coefficients
_Factlist = [1]

def _calc_factlist(nn):
r"""
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]
"""
if nn >= len(_Factlist):
for ii in range(len(_Factlist), nn + 1):
_Factlist.append(_Factlist[ii - 1] * ii)
return _Factlist[:int(nn) + 1]

[docs]def wigner_3j(j_1, j_2, j_3, m_1, m_2, m_3, prec=None): r""" Calculate the Wigner 3j symbol 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 - 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_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:: Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3) =Wigner3j(j_3,j_1,j_2,m_3,m_1,m_2) =Wigner3j(j_2,j_3,j_1,m_2,m_3,m_1) =(-1)^J Wigner3j(j_3,j_2,j_1,m_3,m_2,m_1) =(-1)^J Wigner3j(j_1,j_3,j_2,m_1,m_3,m_2) =(-1)^J Wigner3j(j_2,j_1,j_3,m_2,m_1,m_3) - invariant under space inflection, i.e. .. math:: Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3) =(-1)^J 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 """ if int(j_1 * 2) != j_1 * 2 or int(j_2 * 2) != j_2 * 2 or \ int(j_3 * 2) != j_3 * 2: raise ValueError("j values must be integer or half integer") if int(m_1 * 2) != m_1 * 2 or int(m_2 * 2) != m_2 * 2 or \ int(m_3 * 2) != m_3 * 2: raise ValueError("m values must be integer or half integer") if m_1 + m_2 + m_3 != 0: return 0 prefid = Integer((-1) ** int(j_1 - j_2 - m_3)) m_3 = -m_3 a1 = j_1 + j_2 - j_3 if a1 < 0: return 0 a2 = j_1 - j_2 + j_3 if a2 < 0: return 0 a3 = -j_1 + j_2 + j_3 if a3 < 0: return 0 if (abs(m_1) > j_1) or (abs(m_2) > j_2) or (abs(m_3) > j_3): return 0 maxfact = max(j_1 + j_2 + j_3 + 1, j_1 + abs(m_1), j_2 + abs(m_2), j_3 + abs(m_3)) _calc_factlist(int(maxfact)) argsqrt = Integer(_Factlist[int(j_1 + j_2 - j_3)] * _Factlist[int(j_1 - j_2 + j_3)] * _Factlist[int(-j_1 + j_2 + j_3)] * _Factlist[int(j_1 - m_1)] * _Factlist[int(j_1 + m_1)] * _Factlist[int(j_2 - m_2)] * _Factlist[int(j_2 + m_2)] * _Factlist[int(j_3 - m_3)] * _Factlist[int(j_3 + m_3)]) / \ _Factlist[int(j_1 + j_2 + j_3 + 1)] ressqrt = sqrt(argsqrt) if ressqrt.is_complex: ressqrt = ressqrt.as_real_imag()[0] imin = max(-j_3 + j_1 + m_2, -j_3 + j_2 - m_1, 0) imax = min(j_2 + m_2, j_1 - m_1, j_1 + j_2 - j_3) sumres = 0 for ii in range(int(imin), int(imax) + 1): den = _Factlist[ii] * \ _Factlist[int(ii + j_3 - j_1 - m_2)] * \ _Factlist[int(j_2 + m_2 - ii)] * \ _Factlist[int(j_1 - ii - m_1)] * \ _Factlist[int(ii + j_3 - j_2 + m_1)] * \ _Factlist[int(j_1 + j_2 - j_3 - ii)] sumres = sumres + Integer((-1) ** ii) / den res = ressqrt * sumres * prefid return res
[docs]def clebsch_gordan(j_1, j_2, j_3, m_1, m_2, m_3, prec=None): r""" 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 - 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 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} \; 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 """ res = (-1) ** sympify(j_1 - j_2 + m_3) * sqrt(2 * j_3 + 1) * \ wigner_3j(j_1, j_2, j_3, m_1, m_2, -m_3, prec) return res
def _big_delta_coeff(aa, bb, cc, prec=None): r""" 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) """ if int(aa + bb - cc) != (aa + bb - cc): raise ValueError("j values must be integer or half integer and fulfill the triangle relation") if int(aa + cc - bb) != (aa + cc - bb): raise ValueError("j values must be integer or half integer and fulfill the triangle relation") if int(bb + cc - aa) != (bb + cc - aa): raise ValueError("j values must be integer or half integer and fulfill the triangle relation") if (aa + bb - cc) < 0: return 0 if (aa + cc - bb) < 0: return 0 if (bb + cc - aa) < 0: return 0 maxfact = max(aa + bb - cc, aa + cc - bb, bb + cc - aa, aa + bb + cc + 1) _calc_factlist(maxfact) argsqrt = Integer(_Factlist[int(aa + bb - cc)] * _Factlist[int(aa + cc - bb)] * _Factlist[int(bb + cc - aa)]) / \ Integer(_Factlist[int(aa + bb + cc + 1)]) ressqrt = sqrt(argsqrt) if prec: ressqrt = ressqrt.evalf(prec).as_real_imag()[0] return ressqrt
[docs]def racah(aa, bb, cc, dd, ee, ff, prec=None): r""" 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:: 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 """ prefac = _big_delta_coeff(aa, bb, ee, prec) * \ _big_delta_coeff(cc, dd, ee, prec) * \ _big_delta_coeff(aa, cc, ff, prec) * \ _big_delta_coeff(bb, dd, ff, prec) if prefac == 0: return 0 imin = max(aa + bb + ee, cc + dd + ee, aa + cc + ff, bb + dd + ff) imax = min(aa + bb + cc + dd, aa + dd + ee + ff, bb + cc + ee + ff) maxfact = max(imax + 1, aa + bb + cc + dd, aa + dd + ee + ff, bb + cc + ee + ff) _calc_factlist(maxfact) sumres = 0 for kk in range(imin, imax + 1): den = _Factlist[int(kk - aa - bb - ee)] * \ _Factlist[int(kk - cc - dd - ee)] * \ _Factlist[int(kk - aa - cc - ff)] * \ _Factlist[int(kk - bb - dd - ff)] * \ _Factlist[int(aa + bb + cc + dd - kk)] * \ _Factlist[int(aa + dd + ee + ff - kk)] * \ _Factlist[int(bb + cc + ee + ff - kk)] sumres = sumres + Integer((-1) ** kk * _Factlist[kk + 1]) / den res = prefac * sumres * (-1) ** int(aa + bb + cc + dd) return res
[docs]def wigner_6j(j_1, j_2, j_3, j_4, j_5, j_6, prec=None): r""" Calculate the Wigner 6j symbol 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:: 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:: Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) =Wigner6j(j_3,j_1,j_2,j_6,j_4,j_5) =Wigner6j(j_2,j_3,j_1,j_5,j_6,j_4) =Wigner6j(j_3,j_2,j_1,j_6,j_5,j_4) =Wigner6j(j_1,j_3,j_2,j_4,j_6,j_5) =Wigner6j(j_2,j_1,j_3,j_5,j_4,j_6) - They are invariant under the exchange of the upper and lower arguments in each of any two columns, i.e. .. math:: Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) =Wigner6j(j_1,j_5,j_6,j_4,j_2,j_3) =Wigner6j(j_4,j_2,j_6,j_1,j_5,j_3) =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) """ res = (-1) ** int(j_1 + j_2 + j_4 + j_5) * \ racah(j_1, j_2, j_5, j_4, j_3, j_6, prec) return res
[docs]def wigner_9j(j_1, j_2, j_3, j_4, j_5, j_6, j_7, j_8, j_9, prec=None): r""" Calculate the Wigner 9j symbol 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... 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]_. """ imin = 0 imax = min(j_1 + j_9, j_2 + j_6, j_4 + j_8) sumres = 0 for kk in range(imin, imax + 1): sumres = sumres + (2 * kk + 1) * \ racah(j_1, j_2, j_9, j_6, j_3, kk, prec) * \ racah(j_4, j_6, j_8, j_2, j_5, kk, prec) * \ racah(j_1, j_4, j_9, j_8, j_7, kk, prec) return sumres
[docs]def gaunt(l_1, l_2, l_3, m_1, m_2, m_3, prec=None): r""" Calculate the Gaunt coefficient. The Gaunt coefficient is defined as the integral over three spherical harmonics: .. math:: Y(j_1,j_2,j_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{(2l_1+1)(2l_2+1)(2l_3+1)/(4\pi)} \; Y(j_1,j_2,j_3,0,0,0) \; Y(j_1,j_2,j_3,m_1,m_2,m_3) 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:: Y(j_1,j_2,j_3,m_1,m_2,m_3) =Y(j_3,j_1,j_2,m_3,m_1,m_2) =Y(j_2,j_3,j_1,m_2,m_3,m_1) =Y(j_3,j_2,j_1,m_3,m_2,m_1) =Y(j_1,j_3,j_2,m_1,m_3,m_2) =Y(j_2,j_1,j_3,m_2,m_1,m_3) - invariant under space inflection, i.e. .. math:: Y(j_1,j_2,j_3,m_1,m_2,m_3) =Y(j_1,j_2,j_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. J = 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 """ if int(l_1) != l_1 or int(l_2) != l_2 or int(l_3) != l_3: raise ValueError("l values must be integer") if int(m_1) != m_1 or int(m_2) != m_2 or int(m_3) != m_3: raise ValueError("m values must be integer") bigL = (l_1 + l_2 + l_3) // 2 a1 = l_1 + l_2 - l_3 if a1 < 0: return 0 a2 = l_1 - l_2 + l_3 if a2 < 0: return 0 a3 = -l_1 + l_2 + l_3 if a3 < 0: return 0 if (2 * bigL) % 2 != 0: return 0 if (m_1 + m_2 + m_3) != 0: return 0 if (abs(m_1) > l_1) or (abs(m_2) > l_2) or (abs(m_3) > l_3): return 0 imin = max(-l_3 + l_1 + m_2, -l_3 + l_2 - m_1, 0) imax = min(l_2 + m_2, l_1 - m_1, l_1 + l_2 - l_3) maxfact = max(l_1 + l_2 + l_3 + 1, imax + 1) _calc_factlist(maxfact) argsqrt = (2 * l_1 + 1) * (2 * l_2 + 1) * (2 * l_3 + 1) * \ _Factlist[l_1 - m_1] * _Factlist[l_1 + m_1] * _Factlist[l_2 - m_2] * \ _Factlist[l_2 + m_2] * _Factlist[l_3 - m_3] * _Factlist[l_3 + m_3] / \ (4*pi) ressqrt = sqrt(argsqrt) prefac = Integer(_Factlist[bigL] * _Factlist[l_2 - l_1 + l_3] * _Factlist[l_1 - l_2 + l_3] * _Factlist[l_1 + l_2 - l_3])/ \ _Factlist[2 * bigL + 1]/ \ (_Factlist[bigL - l_1] * _Factlist[bigL - l_2] * _Factlist[bigL - l_3]) sumres = 0 for ii in range(imin, imax + 1): den = _Factlist[ii] * _Factlist[ii + l_3 - l_1 - m_2] * \ _Factlist[l_2 + m_2 - ii] * _Factlist[l_1 - ii - m_1] * \ _Factlist[ii + l_3 - l_2 + m_1] * _Factlist[l_1 + l_2 - l_3 - ii] sumres = sumres + Integer((-1) ** ii) / den res = ressqrt * prefac * sumres * (-1) ** (bigL + l_3 + m_1 - m_2) if prec is not None: res = res.n(prec) return res