def getRadialMatrixElementSemiClassical(self, n, l, s, j, n1, l1, s1, j1):
#get the effective principal number of both states
nu = n - self.getQuantumDefect(n, l, s, j)
nu1 = n1 - self.getQuantumDefect(n1, l1, s1, j1)
#get the parameters required to calculate the sum
l_c = (l + l1 + 1.) / 2.
nu_c = sqrt(nu * nu1)
delta_nu = nu - nu1
delta_l = l1 - l
gamma = (delta_l * l_c) / nu_c
g0 = (1. /
(3. * delta_nu)) * (mpmath.angerj(delta_nu - 1., -delta_nu) -
mpmath.angerj(delta_nu + 1, -delta_nu))
g1 = -(1. /
(3. * delta_nu)) * (mpmath.angerj(delta_nu - 1., -delta_nu) +
mpmath.angerj(delta_nu + 1, -delta_nu))
g2 = g0 - mpmath.sin(pi * delta_nu) / (pi * delta_nu)
g3 = (delta_nu / 2.) * g0 + g1
radial_ME = (3 / 2) * nu_c**2 * (1 - (l_c / nu_c)**(2))**0.5 * (
g0 + gamma * g1 + gamma**2 * g2 + gamma**3 * g3)
return radial_ME
def ExactIntegrationOfSinus(t, a=None, b=None):
with precision(300):
if a == None and b == None:
return 0.5 * math.pi * math.sqrt(t) * (mpmath.angerj(0.5, t) -
mpmath.angerj(-0.5, t))
elif a == None and b != None:
a = 0
elif b == None:
b = t
mpmath.mp.dps = 50
mpmath.mp.pretty = True
pi = mpmath.mp.pi
pi = +pi
fcos = mpmath.fresnelc
fsin = mpmath.fresnels
arg_b = mpmath.sqrt(2 * (t - b) / pi)
if a == "MinusInfinity":
return mpmath.sqrt(
0.5 * mpmath.mp.pi) * (-2 * mpmath.sin(t) * fcos(arg_b) +
2 * mpmath.cos(t) * fsin(arg_b) +
mpmath.sin(t) - mpmath.cos(t))
else:
arg_a = mpmath.sqrt(2 * (t - a) / pi)
return mpmath.sqrt(2 * mpmath.mp.pi) * (
(fsin(arg_b) - fsin(arg_a)) * mpmath.cos(t) +
(fcos(arg_a) - fcos(arg_b)) * mpmath.sin(t))
def ExactIntegrationOfSinusKernel(t, a = None, b = None):
with precision(300):
if a == None and b == None:
return 0.5 * math.pi * math.sqrt(t) * (mpmath.angerj(0.5, t) - mpmath.angerj(-0.5, t))
if a == None and b != None:
a = t
elif b == None:
b = 0.
mpmath.mp.pretty = True
pi = mpmath.mp.pi
pi = +pi
fcos = mpmath.fresnelc
fsin = mpmath.fresnels
arg_a = mpmath.sqrt(2 * (t - a) / mpmath.mp.pi)
arg_b = mpmath.sqrt(2 * (t - b) / mpmath.mp.pi)
return mpmath.sqrt(2 * mpmath.mp.pi) * ((fsin(arg_b) - fsin(arg_a)) * mpmath.cos(t) + (fcos(arg_a) - fcos(arg_b)) * mpmath.sin(t))