Meshgrid demo with Spherical Harmonics

Meshgrid : Spherical Harmonics

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
sns.set()

Example : Spherical Harmonics

$$Y(\theta,\phi) : \langle \theta,\phi|l,m \rangle $$

value of l	value of m	Spherical Harmonics in position space
l=0	m=0	$$ Y_0^0(\theta,\phi)=\frac{1}{2}\sqrt{\frac{1}{\pi}}$$
l=1	m=-1	$$ Y_1^{-1}(\theta, \phi) = \frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{-i\theta} \sin(\phi) $$
l=1	m=0	$$ Y_1^0(\theta, \phi) = \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos(\phi) $$
l=1	m=1	$$ Y_1^1(\theta, \phi) = -\frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{i\theta} \sin(\phi) $$

import scipy.special as sp
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D

phis,thetas = np.mgrid[0:2*np.pi:200j, 0:np.pi:100j]

Example 1 : $$Y_0^0(\theta, \phi) = \frac{1}{2} \sqrt{\frac{1}{\pi}}$$

l = 0
m = 0   

RG = np.abs(sp.sph_harm(m,l,phis,thetas)) 

'''spherical to cartesian'''
XG = RG * np.sin(thetas) * np.cos(phis)
YG = RG * np.sin(thetas) * np.sin(phis)
ZG = RG * np.cos(thetas)

'''normalization with maximum value'''
NG = RG/RG.max()

'''plot figure'''
fig = plt.figure(figsize = [8,8])
ax = fig.gca(projection='3d')

surf = ax.plot_surface(XG, YG, ZG,\
                       cmap=cm.coolwarm,\
                       linewidth=0)

plt.title(r'$|Y^1_ 0|$', fontsize=20)
plt.show()

png

Example 2: $$ Y_2^1(\theta, \phi)$$

l = 2
m = 1 

RG = np.abs(sp.sph_harm(m,l,phis,thetas)) 

'''spherical to cartesian'''
XG = RG * np.sin(thetas) * np.cos(phis)
YG = RG * np.sin(thetas) * np.sin(phis)
ZG = RG * np.cos(thetas)

'''normalization with maximum value'''
NG = RG/RG.max()

'''plot figure'''
fig = plt.figure(figsize = [8,8])
ax = fig.gca(projection='3d')

surf = ax.plot_surface(XG, YG, ZG,\
                       cmap=cm.coolwarm,\
                       linewidth=0)

plt.title(r'$|Y^1_ 0|$', fontsize=20)
plt.show()

png

Example 3: $$Y_4^2(\theta, \phi)$$ Real part

l = 4  
m = 2  

RG = np.abs(sp.sph_harm(m,l,phis,thetas).real) 

'''spherical to cartesian'''
XG = RG * np.sin(thetas) * np.cos(phis)
YG = RG * np.sin(thetas) * np.sin(phis)
ZG = RG * np.cos(thetas)

'''normalization with maximum value'''
NG = RG/RG.max()

'''plot figure'''
fig = plt.figure(figsize = [8,8])
ax = fig.gca(projection='3d')

surf = ax.plot_surface(XG, YG, ZG,\
                       cmap=cm.coolwarm,\
                       linewidth=0)

plt.title(r'$|Y^4_ 2|$', fontsize=20)
plt.show()

png

Example 4 : $$Y_5^4(\theta, \phi)$$ Imaginary Part

l = 5  
m = 4   

RG = np.abs(sp.sph_harm(m,l,phis,thetas).imag) 

'''spherical to cartesian'''
XG = RG * np.sin(thetas) * np.cos(phis)
YG = RG * np.sin(thetas) * np.sin(phis)
ZG = RG * np.cos(thetas)

'''normalization with maximum value'''
NG = RG/RG.max()

'''plot figure'''
fig = plt.figure(figsize = [8,8])
ax = fig.gca(projection='3d')

surf = ax.plot_surface(XG, YG, ZG,\
                       cmap=cm.coolwarm,\
                       linewidth=0)

plt.title(r'$|Y^5_ 4|$', fontsize=20)
plt.show()

png