def LandauLevelSpinor(self, B , n , x , y ,type=1):
# Symmetric Gauge
def energy(n):
return np.sqrt( (self.mass*self.c**2)**2 + 2*B*self.c*self.hBar*n )
K = B*( (self.X-x)**2 + (self.Y-y)**2)/( 4.*self.c*self.hBar )
psi1 = np.exp(-K)*( energy(n) + self.mass*self.c**2 )*laguerre(n)( 2*K )
psi3 = np.exp(-K)*( energy(n) - self.mass*self.c**2 )*laguerre(n)( 2*K )
if n>0:
psi2 = 1j*np.exp(-K)*( self.X-x + 1j*(self.Y-y) )*genlaguerre(n-1,1)( 2*K )
else:
psi2 = 0.*K
psi4 = psi2
if type==1:
spinor = np.array([ psi1 , 0*psi2 , 0*psi3 , psi4 ])
elif type ==2:
spinor = np.array([ 0*psi1 , psi2 , psi3 , 0*psi4 ])
else :
print 'Error: type spinor must be 1 or 2'
norm = self.Norm(spinor)
spinor /= norm
return spinor
def LandauLevelSpinor(self, B , n , x , y ,type=1):
# Symmetric Gauge
def energy(n):
return np.sqrt( (self.mass*self.c**2)**2 + 2*B*self.c*self.hBar*n )
K = B*( (self.X-x)**2 + (self.Y-y)**2)/( 4.*self.c*self.hBar )
psi1 = np.exp(-K)*( energy(n) + self.mass*self.c**2 )*laguerre(n)( 2*K )
psi3 = np.exp(-K)*( energy(n) - self.mass*self.c**2 )*laguerre(n)( 2*K )
if n>0:
psi2 = 1j*np.exp(-K)*( self.X-x + 1j*(self.Y-y) )*genlaguerre(n-1,1)( 2*K )
else:
psi2 = 0.*K
psi4 = psi2
if type==1:
spinor = np.array([ psi1 , 0*psi2 , 0*psi3 , psi4 ])
elif type ==2:
spinor = np.array([ 0*psi1 , psi2 , psi3 , 0*psi4 ])
else :
print 'Error: type spinor must be 1 or 2'
norm = self.Norm(spinor)
spinor /= norm
return spinor
def test_wigner_fock():
"wigner: test wigner function calculation for Fock states"
xvec = linspace(-5.0, 5.0, 100)
yvec = xvec
X, Y = meshgrid(xvec, yvec)
a = X + 1j * Y # consistent with g=2 option to wigner function
dx = xvec[1] - xvec[0]
dy = yvec[1] - yvec[0]
N = 15
for n in [2, 3, 4, 5, 6]:
psi = fock(N, n)
# calculate the wigner function using qutip and analytic formula
W_qutip = wigner(psi, xvec, yvec, g=2)
W_analytic = 2 / pi * (-1)**n * exp(-2 * abs(a)**2) * polyval(
laguerre(n), 4 * abs(a)**2)
# check difference
assert_(sum(abs(W_qutip - W_analytic)) < 1e-4)
# check normalization
assert_(sum(W_qutip) * dx * dy - 1.0 < 1e-8)
assert_(sum(W_analytic) * dx * dy - 1.0 < 1e-8)
def test_wigner_fock():
"wigner: test wigner function calculation for Fock states"
xvec = linspace(-5.0, 5.0, 100)
yvec = xvec
X,Y = meshgrid(xvec, yvec)
a = X + 1j * Y # consistent with g=2 option to wigner function
dx = xvec[1]-xvec[0]
dy = yvec[1]-yvec[0]
N = 15
for n in [2,3,4,5,6]:
psi = fock(N, n)
# calculate the wigner function using qutip and analytic formula
W_qutip = wigner(psi, xvec, yvec, g=2)
W_analytic = 2/pi * (-1)**n * exp(-2*abs(a)**2) * polyval(laguerre(n), 4*abs(a)**2)
# check difference
assert_(sum(abs(W_qutip - W_analytic)) < 1e-4)
# check normalization
assert_(sum(W_qutip) * dx * dy - 1.0 < 1e-8)
assert_(sum(W_analytic) * dx * dy - 1.0 < 1e-8)
def W(x, y):
#Coherent state
#return 2/np.pi* np.exp(-2*(x-x0)**2-2*(y-y0)**2)
#Number state
L = laguerre(n)
return 2 / np.pi * (-1)**n * L(4 * (x**2 + y**2)) * np.exp(-2 *
(x**2 + y**2))
def R_func(r, n, l):
N = (2 / (n * a0))**(3 / 2) * np.sqrt(
(np.math.factorial(n - l - 1)) / (2 * n *
((np.math.factorial(n + l)))**3))
L = laguerre(n + l)
L = L / np.abs(L[n + l])
assoc_L = derivative(L, 2 * r / (n * a0), n=2 * l + 1, order=2 * l + 3)
R = -N * (2 * r / (n * a0))**l * np.exp(-r / (n * a0)) * assoc_L
return R
# -*- coding: utf-8 -*-
from matplotlib.pyplot import *
from numpy import *
from scipy.special import laguerre
colores=['blue','red','brown','purple','black']
dasheses=[[],[5,2],[5,5],[5,2,2,2],[2,2]]
x = linspace(-5,20,1000)
fig = figure(figsize=(8,6))
for n in range(5):
plot(x,polyval(laguerre(n),x),colores[n], dashes=dasheses[n],label=r'$L_{%d}(x)$'%n, linewidth=2)
ylim(-10,20)
xlabel('$x$',fontsize=15)
ylabel(r'$L_n(x)$',fontsize=15)
legend(loc='best',fontsize=13)
grid()
savefig('../figs/fig-Laguerre.pdf')
import numpy as np
from numpy import exp, sqrt
from scipy.special import laguerre, gamma
from matplotlib import pyplot as plt
from matplotlib import cm
n = 10
x = np.linspace(0.01, 30, 500)
for i in range(n):
plt.plot(x, exp(- x / i) * laguerre(i)(2 * x / i) / i**2, color = cm.viridis(i /n))
plt.show()
def laguerreTableInternal(eta):
logger = logging.getLogger(__name__)
eta2 = pow(eta, 2)
logger.info( "Calculating Laguerre Table for eta^2={0}".format(eta2) )
laguerreTable = array([ laguerre(n)(eta2) for n in range(200) ])
return laguerreTable
def laguerreTableInternal(eta):
logger = logging.getLogger(__name__)
eta2 = pow(eta, 2)
logger.info("Calculating Laguerre Table for eta^2={0}".format(eta2))
laguerreTable = array([laguerre(n)(eta2) for n in range(200)])
return laguerreTable
