def test_wigner():
def tn(a, b):
return abs((a - b).n(64) < S('1e-64'))
assert tn(wigner_9j(1,1,1, 1,1,1, 1,1,0 ,prec=64), S(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), -S(12219)/965770)
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), S(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), -S(12219)/965770)
def safe_wigner_9j(*args):
""" Wigner 9j symbol, same arguments as in Avoid the ValueError whenever the
arguments don't fulfill the triangle relation. """
try:
return float(wigner_9j(*args, prec=None))
except ValueError or AttributeError:
return 0
def dump_w9j(tjmax):
# note: sympy's wigner_9j is buggy for half-integer arguments
# (as of 0.7.6.1)
import sympy.physics.wigner as spw
for write in dump_output("symw9j", tjmax):
for tjs in iter_9tjs(tjmax):
z = spw.wigner_9j(*(Fraction(tj, 2) for tj in tjs))
write("\t".join(map(str, tjs + (show_signedsqrtfrac(z),))) + "\n")
def dump_w9j(tjmax):
# note: sympy's wigner_9j is buggy for half-integer arguments
# (as of 0.7.6.1)
import sympy.physics.wigner as spw
for write in dump_output("symw9j", tjmax):
for tjs in iter_9tjs(tjmax):
z = spw.wigner_9j(*(Fraction(tj, 2) for tj in tjs))
write("\t".join(map(str, tjs + (show_signedsqrtfrac(z), ))) + "\n")
def Ysigma(l1, j1, l2, j2, l, s, k):
"""
< l1, j1 || [Y_l sigma_s]_k || l2, j2 >
Note: all inputs are not not doubled
"""
sfact = np.sqrt(2.0)
if (s == 1): sfact = np.sqrt(6.0)
return np.sqrt(float(
(2 * j1 + 1) * (2 * j2 + 1) *
(2 * k + 1))) * N(wigner_9j(l1, 0.5, j1, l2, 0.5, j2, l, s, k,
prec=8)) * Yl_red(l1, l2, l) * sfact
def compute_matrix_coeff(i):
# output submatrix
tmp_out = np.zeros((len(ell_1), len(ell_1)))
# i is first matrix index
L_1, L_2, L_3 = ell_1[i], ell_2[i], ell_3[i]
pref_1 = np.sqrt((2. * L_1 + 1.) * (2. * L_2 + 1.) * (2. * L_3 + 1.))
for j in range(len(ell_1)):
# j is second matrix index
Lpp_1, Lpp_2, Lpp_3 = ell_1[j], ell_2[j], ell_3[j]
pref_2 = pref_1 * np.sqrt(
(2. * Lpp_1 + 1.) * (2. * Lpp_2 + 1.) * (2. * Lpp_3 + 1.))
# add phase
pref_2 *= (-1.)**(Lpp_1 + Lpp_2 + Lpp_3)
for k in range(len(ell_1)):
# k indexes inner Lambda' term
Lp_1, Lp_2, Lp_3 = ell_1[k], ell_2[k], ell_3[k]
# Compute prefactor
pref = pref_2 * np.sqrt(
(2. * Lp_1 + 1.) * (2. * Lp_2 + 1.) *
(2. * Lp_3 + 1.)) / (4. * np.pi)**(3. / 2.)
# Compute 3j couplings
three_j_piece = np.float64(wigner_3j(L_1, Lp_1, Lpp_1, 0, 0,
0))
if three_j_piece == 0: continue
three_j_piece *= np.float64(
wigner_3j(L_2, Lp_2, Lpp_2, 0, 0, 0))
if three_j_piece == 0: continue
three_j_piece *= np.float(wigner_3j(L_3, Lp_3, Lpp_3, 0, 0, 0))
if three_j_piece == 0: continue
# Compute the 9j component
nine_j_piece = np.float64(
wigner_9j(L_1,
Lp_1,
Lpp_1,
L_2,
Lp_2,
Lpp_2,
L_3,
Lp_3,
Lpp_3,
prec=8))
if nine_j_piece == 0: continue
tmp_out[j, k] = pref * three_j_piece * nine_j_piece
return tmp_out
def doit(self, **hints):
if self.is_symbolic:
raise ValueError("Coefficients must be numerical")
return wigner_9j(
self.j1,
self.j2,
self.j12,
self.j3,
self.j4,
self.j34,
self.j13,
self.j24,
self.j,
)
def tensor_product(a, b, j, k1, k2, k, a1, b1, j1, n=True, precision=None):
"""
If you have a matrix element of a product of two tensor operators
of rank k1 and k2, this product operator has a spherical tensor
operator component of rank k.
This function describes the reduced matrix element:
<J,A,B||(T^k1(A) T^k2(B))^k||J1,A1,B1>
see Brown and Carrington 5.169
"""
output = (sqrt((2 * j + 1) * (2 * k + 1) * (2 * j1 + 1)) *
wigner_9j(a1, b1, j1, k1, k2, k, a, b, j, precision=precision))
return formated_output(output, n=n)
def tensor_product(a, b, j,
k1, k2, k,
a1, b1, j1,
n=True, precision=None):
"""
If you have a matrix element of a product of two tensor operators
of rank k1 and k2, this product operator has a spherical tensor
operator component of rank k.
This function describes the reduced matrix element:
<J,A,B||(T^k1(A) T^k2(B))^k||J1,A1,B1>
see Brown and Carrington 5.169
"""
output = (sqrt((2 * j + 1) * (2 * k + 1) * (2 * j1 + 1)) *
wigner_9j(a1, b1, j1,
k1, k2, k,
a, b, j,
precision=precision))
return formated_output(output, n=n)
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), S(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), -S(12219) / 965770)
# regression test for #8747
half = Rational(1, 2)
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_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), S(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), -S(12219)/965770)
# regression test for #8747
half = Rational(1, 2)
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 doit(self, **hints):
if self.is_symbolic:
raise ValueError("Coefficients must be numerical")
return wigner_9j(self.j1, self.j2, self.j12, self.j3, self.j4, self.j34, self.j13, self.j24, self.j)
for k, y in enumerate(it.repeat((jpair[0], jpair[1]),
len(Js))):
jjj_wig3j = wigner_3j(y[0], y[1], Js[k], 0, 0, 0).evalf()
maxp = max(x[0] + y[0], x[1] + y[1])
for ppair in it.product(np.arange(maxp + 1), repeat=2):
Ps = return_Ls(ppair[0], ppair[1])
for m, z in enumerate(
it.repeat((ppair[0], ppair[1]), len(Ps))):
wig9j = wigner_9j(x[0],
y[0],
z[0],
x[1],
y[1],
z[1],
Ls[i],
Js[k],
Ps[m],
prec=64)
cleb_p0 = clebsch_gordan(z[0], x[0], y[0], 0, 0,
0).evalf()
cleb_p1 = clebsch_gordan(z[1], x[1], y[1], 0, 0,
0).evalf()
cleb_P0 = clebsch_gordan(Ps[m], Ls[i], Js[k], 0, 0,
0).evalf()
if np.abs(ell_wig3j) > 0.0 and np.abs(
jjj_wig3j) > 0.0 and np.abs(wig9j) > 0.0:
if np.abs(cleb_p0) > 0.0 and np.abs(
def non_antisimetrized_BB_ME(bra, ket, parameters):
# Passing of parameters
n_a, n_b = bra.n1, bra.n2
n_c, n_d = ket.n1, ket.n2
l_a, l_b = bra.l1, bra.l2
l_c, l_d = ket.l1, ket.l2
j_a, j_b = bra.j1, bra.j2
j_c, j_d = ket.j1, ket.j2
J, T = bra.J, bra.T
# Adjust the oscillator lenght to the reducced mass of the proton
b_param = parameters[0][0] * np.sqrt(2)
Sum = 0.
L_max = min((l_a + l_b), (l_c + l_d))
L_min = max(abs(l_a - l_b), abs(l_c - l_d))
for L in range(L_min, L_max + 1):
for S in range(0, 2):
try:
# j atribute are defined as 2*j
w9j_bra = wigner_9j(l_a,
1 / 2,
j_a / 2,
l_b,
1 / 2,
j_b / 2,
L,
S,
J,
prec=None) #.evalf()
except ValueError:
w9j_bra = 0
if (abs(w9j_bra) > 1.e-15):
try:
w9j_ket = wigner_9j(l_c,
1 / 2,
j_c / 2,
l_d,
1 / 2,
j_d / 2,
L,
S,
J,
prec=None) #.evalf()
except ValueError:
w9j_ket = 0
if (abs(w9j_ket) > 1.e-15):
# j atribute are defined as 2*j
recoupling = ((2 * S + 1) * (2 * L + 1) * np.sqrt(
(j_a + 1) * (j_b + 1) * (j_c + 1) * (j_d + 1)) *
w9j_bra * w9j_ket)
# Sum of gaussians and projection operators
for i in range(2):
mu_param = parameters[i][1]
# Radial Part for Gauss Integral (L == lambda)
Radial = Moshinsky_Transformation_ME(
n_a, l_a, n_b, l_b, n_c, l_c, n_d, l_d, L, b_param,
mu_param)
# print('Radial['+ str(i)+ ']: '+str(Radial))
# Exchange Part
Exchange_Energy = (
parameters[i][2] + # Wigner
(parameters[i][3] * ((-1)**(L))) + # Majorana
(parameters[i][4] * ((-1)**(S))) + # Barlett
(parameters[i][5] * ((-1)**(T)))) # Heisenberg
# print('Exchange['+ str(i)+ ']: ',Exchange_Energy, ' couping=',recoupling)
# Add up
Sum += recoupling * Radial * Exchange_Energy
return Sum
j8 = map(float, data[:, 7])
j9 = map(float, data[:, 8])
W_9j = map(float, data[:, 9])
errors, non_zero = 0, 0
tol = 1e-5
Erroneos, No_erroneos = [], []
for i in range(len(j1)):
w_9j, j_1, j_2, j_3, j_4, j_5, j_6, j_7, j_8, j_9 = \
W_9j[i], j1[i], j2[i], j3[i], j4[i], j5[i], j6[i], j7[i], j8[i], j9[i]
try:
w9j_analitic = wigner_9j(j_1,
j_2,
j_3,
j_4,
j_5,
j_6,
j_7,
j_8,
j_9,
prec=None)
except ValueError:
w9j_analitic = 0
if abs(w_9j - w9j_analitic) > tol:
errors += 1
non_zero += 1
# print(j1, j2, j, m1, m2, m,"cg=",cg," // cg_an=",cg_analitic," //Error")
Erroneos.append([[j_1, j_2, j_3, j_4, j_5, j_6, j_7, j_8, j_9],
w_9j, w9j_analitic])
raw_input("press any key")
# else:
# print(j1, j2, j, m1, m2, m,"cg=",cg," // cg_an=",cg_analitic)
def wig9j(j1, j2, j3, l1, l2, l3, n1, n2, n3):
"""
This function redefines the wig9jj in terms of things that I like:
"""
return float(wigner_9j(j1, j2, j3, l1, l2, l3, n1, n2, n3))