from sympy.core.numbers import (I, pi) from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import exp from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (cos, sin) from sympy.functions.special.spherical_harmonics import Ynm from sympy.matrices.dense import Matrix from sympy.physics.wigner import (clebsch_gordan, wigner_9j, wigner_6j, gaunt, racah, dot_rot_grad_Ynm, wigner_3j, wigner_d_small, wigner_d) from sympy.core.numbers import Rational # for test cases, refer : https://en.wikipedia.org/wiki/Table_of_Clebsch%E2%80%93Gordan_coefficients def test_clebsch_gordan_docs(): assert clebsch_gordan(Rational(3, 2), S.Half, 2, Rational(3, 2), S.Half, 2) == 1 assert clebsch_gordan(Rational(3, 2), S.Half, 1, Rational(3, 2), Rational(-1, 2), 1) == sqrt(3)/2 assert clebsch_gordan(Rational(3, 2), S.Half, 1, Rational(-1, 2), S.Half, 0) == -sqrt(2)/2 def test_clebsch_gordan1(): j_1 = S.Half j_2 = S.Half m = 1 j = 1 m_1 = S.Half m_2 = S.Half assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1 j_1 = S.Half j_2 = S.Half m = -1 j = 1 m_1 = Rational(-1, 2) m_2 = Rational(-1, 2) assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1 j_1 = S.Half j_2 = S.Half m = 0 j = 1 m_1 = S.Half m_2 = S.Half assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 0 j_1 = S.Half j_2 = S.Half m = 0 j = 1 m_1 = S.Half m_2 = Rational(-1, 2) assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == sqrt(2)/2 j_1 = S.Half j_2 = S.Half m = 0 j = 0 m_1 = S.Half m_2 = Rational(-1, 2) assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == sqrt(2)/2 j_1 = S.Half j_2 = S.Half m = 0 j = 1 m_1 = Rational(-1, 2) m_2 = S.Half assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == sqrt(2)/2 j_1 = S.Half j_2 = S.Half m = 0 j = 0 m_1 = Rational(-1, 2) m_2 = S.Half assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == -sqrt(2)/2 def test_clebsch_gordan2(): j_1 = S.One j_2 = S.Half m = Rational(3, 2) j = Rational(3, 2) m_1 = 1 m_2 = S.Half assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1 j_1 = S.One j_2 = S.Half m = S.Half j = Rational(3, 2) m_1 = 1 m_2 = Rational(-1, 2) assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1/sqrt(3) j_1 = S.One j_2 = S.Half m = S.Half j = S.Half m_1 = 1 m_2 = Rational(-1, 2) assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == sqrt(2)/sqrt(3) j_1 = S.One j_2 = S.Half m = S.Half j = S.Half m_1 = 0 m_2 = S.Half assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == -1/sqrt(3) j_1 = S.One j_2 = S.Half m = S.Half j = Rational(3, 2) m_1 = 0 m_2 = S.Half assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == sqrt(2)/sqrt(3) j_1 = S.One j_2 = S.One m = S(2) j = S(2) m_1 = 1 m_2 = 1 assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1 j_1 = S.One j_2 = S.One m = 1 j = S(2) m_1 = 1 m_2 = 0 assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1/sqrt(2) j_1 = S.One j_2 = S.One m = 1 j = S(2) m_1 = 0 m_2 = 1 assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1/sqrt(2) j_1 = S.One j_2 = S.One m = 1 j = 1 m_1 = 1 m_2 = 0 assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1/sqrt(2) j_1 = S.One j_2 = S.One m = 1 j = 1 m_1 = 0 m_2 = 1 assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == -1/sqrt(2) def test_clebsch_gordan3(): j_1 = Rational(3, 2) j_2 = Rational(3, 2) m = S(3) j = S(3) m_1 = Rational(3, 2) m_2 = Rational(3, 2) assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1 j_1 = Rational(3, 2) j_2 = Rational(3, 2) m = S(2) j = S(2) m_1 = Rational(3, 2) m_2 = S.Half assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1/sqrt(2) j_1 = Rational(3, 2) j_2 = Rational(3, 2) m = S(2) j = S(3) m_1 = Rational(3, 2) m_2 = S.Half assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1/sqrt(2) def test_clebsch_gordan4(): j_1 = S(2) j_2 = S(2) m = S(4) j = S(4) m_1 = S(2) m_2 = S(2) assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1 j_1 = S(2) j_2 = S(2) m = S(3) j = S(3) m_1 = S(2) m_2 = 1 assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1/sqrt(2) j_1 = S(2) j_2 = S(2) m = S(2) j = S(3) m_1 = 1 m_2 = 1 assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 0 def test_clebsch_gordan5(): j_1 = Rational(5, 2) j_2 = S.One m = Rational(7, 2) j = Rational(7, 2) m_1 = Rational(5, 2) m_2 = 1 assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1 j_1 = Rational(5, 2) j_2 = S.One m = Rational(5, 2) j = Rational(5, 2) m_1 = Rational(5, 2) m_2 = 0 assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == sqrt(5)/sqrt(7) j_1 = Rational(5, 2) j_2 = S.One m = Rational(3, 2) j = Rational(3, 2) m_1 = S.Half m_2 = 1 assert clebsch_gordan(j_1, j_2, j, m_1, m_2, m) == 1/sqrt(15) def test_wigner(): def tn(a, b): return (a - b).n(64) < S('1e-64') assert tn(wigner_9j(1, 1, 1, 1, 1, 1, 1, 1, 0, prec=64), Rational(1, 18)) assert wigner_9j(3, 3, 2, 3, 3, 2, 3, 3, 2) == 3221*sqrt( 70)/(246960*sqrt(105)) - 365/(3528*sqrt(70)*sqrt(105)) assert wigner_6j(5, 5, 5, 5, 5, 5) == Rational(1, 52) assert tn(wigner_6j(8, 8, 8, 8, 8, 8, prec=64), Rational(-12219, 965770)) # regression test for #8747 half = S.Half assert wigner_9j(0, 0, 0, 0, half, half, 0, half, half) == half assert (wigner_9j(3, 5, 4, 7 * half, 5 * half, 4, 9 * half, 9 * half, 0) == -sqrt(Rational(361, 205821000))) assert (wigner_9j(1, 4, 3, 5 * half, 4, 5 * half, 5 * half, 2, 7 * half) == -sqrt(Rational(3971, 373403520))) assert (wigner_9j(4, 9 * half, 5 * half, 2, 4, 4, 5, 7 * half, 7 * half) == -sqrt(Rational(3481, 5042614500))) def test_gaunt(): def tn(a, b): return (a - b).n(64) < S('1e-64') assert gaunt(1, 0, 1, 1, 0, -1) == -1/(2*sqrt(pi)) assert isinstance(gaunt(1, 1, 0, -1, 1, 0).args[0], Rational) assert isinstance(gaunt(0, 1, 1, 0, -1, 1).args[0], Rational) assert tn(gaunt( 10, 10, 12, 9, 3, -12, prec=64), (Rational(-98, 62031)) * sqrt(6279)/sqrt(pi)) def gaunt_ref(l1, l2, l3, m1, m2, m3): return ( sqrt((2 * l1 + 1) * (2 * l2 + 1) * (2 * l3 + 1) / (4 * pi)) * wigner_3j(l1, l2, l3, 0, 0, 0) * wigner_3j(l1, l2, l3, m1, m2, m3) ) threshold = 1e-10 l_max = 3 l3_max = 24 for l1 in range(l_max + 1): for l2 in range(l_max + 1): for l3 in range(l3_max + 1): for m1 in range(-l1, l1 + 1): for m2 in range(-l2, l2 + 1): for m3 in range(-l3, l3 + 1): args = l1, l2, l3, m1, m2, m3 g = gaunt(*args) g0 = gaunt_ref(*args) assert abs(g - g0) < threshold if m1 + m2 + m3 != 0: assert abs(g) < threshold if (l1 + l2 + l3) % 2: assert abs(g) < threshold def test_racah(): assert racah(3,3,3,3,3,3) == Rational(-1,14) assert racah(2,2,2,2,2,2) == Rational(-3,70) assert racah(7,8,7,1,7,7, prec=4).is_Float assert racah(5.5,7.5,9.5,6.5,8,9) == -719*sqrt(598)/1158924 assert abs(racah(5.5,7.5,9.5,6.5,8,9, prec=4) - (-0.01517)) < S('1e-4') def test_dot_rota_grad_SH(): theta, phi = symbols("theta phi") assert dot_rot_grad_Ynm(1, 1, 1, 1, 1, 0) != \ sqrt(30)*Ynm(2, 2, 1, 0)/(10*sqrt(pi)) assert dot_rot_grad_Ynm(1, 1, 1, 1, 1, 0).doit() == \ sqrt(30)*Ynm(2, 2, 1, 0)/(10*sqrt(pi)) assert dot_rot_grad_Ynm(1, 5, 1, 1, 1, 2) != \ 0 assert dot_rot_grad_Ynm(1, 5, 1, 1, 1, 2).doit() == \ 0 assert dot_rot_grad_Ynm(3, 3, 3, 3, theta, phi).doit() == \ 15*sqrt(3003)*Ynm(6, 6, theta, phi)/(143*sqrt(pi)) assert dot_rot_grad_Ynm(3, 3, 1, 1, theta, phi).doit() == \ sqrt(3)*Ynm(4, 4, theta, phi)/sqrt(pi) assert dot_rot_grad_Ynm(3, 2, 2, 0, theta, phi).doit() == \ 3*sqrt(55)*Ynm(5, 2, theta, phi)/(11*sqrt(pi)) assert dot_rot_grad_Ynm(3, 2, 3, 2, theta, phi).doit().expand() == \ -sqrt(70)*Ynm(4, 4, theta, phi)/(11*sqrt(pi)) + \ 45*sqrt(182)*Ynm(6, 4, theta, phi)/(143*sqrt(pi)) def test_wigner_d(): half = S(1)/2 alpha, beta, gamma = symbols("alpha, beta, gamma", real=True) d = wigner_d_small(half, beta).subs({beta: pi/2}) d_ = Matrix([[1, 1], [-1, 1]])/sqrt(2) assert d == d_ D = wigner_d(half, alpha, beta, gamma) assert D[0, 0] == exp(I*alpha/2)*exp(I*gamma/2)*cos(beta/2) assert D[0, 1] == exp(I*alpha/2)*exp(-I*gamma/2)*sin(beta/2) assert D[1, 0] == -exp(-I*alpha/2)*exp(I*gamma/2)*sin(beta/2) assert D[1, 1] == exp(-I*alpha/2)*exp(-I*gamma/2)*cos(beta/2)