Spherical Harmonics Visualization (Python Notebook)

Source: https://github.com/DalInar/schroding...lization.ipynb

The Legendre Polynomials

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import numpy as np

import matplotlib.pyplot as plt

from scipy.special import lpmv

​

ls = [0,1,2,3]

x=np.linspace(-1,1,100)

​

plt.figure()

​

for l in ls:

    plt.plot(x,lpmv(0,l,x),label=r'$l=$'+str(l))

plt.title(r'Legendre Polynomials, $P_l(x)$')

plt.xlabel(r'$x$')

plt.ylabel(r'$P_l(x)$')

plt.legend()

plt.grid()

plt.show()

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The Legendre Polynomials on a polar plot

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import numpy as np

import matplotlib.pyplot as plt

from scipy.special import lpmv

​

ls = [0,1,2,3]

​

thetas = np.linspace(0,2*np.pi,200)

​

plt.figure(figsize=(8,8))

for i in range(0, len(ls)):

    l=ls[i]

    r = lpmv(0,l,np.cos(thetas))

    plt.polar(thetas, abs(r),label=r'$l=$'+str(l))

plt.title(r'Associated Legendre Polynomials, $||P_l(x)||$')

plt.ylabel(r'$||P_l(x)||$')

plt.legend()

plt.show()

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The Associated Polynomials  

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import numpy as np

import matplotlib.pyplot as plt

from scipy.special import lpmv

​

ls = [0,1,1,1,2]

ms = [0,-1,0,1,1]

​

x=np.linspace(-1,1,100)

​

plt.figure()

​

for i in range(0, len(ls)):

    l=ls[i]

    m=ms[i]

    plt.plot(x,lpmv(m,l,x),label=r'$l=$'+str(l))

plt.title(r'Associated Legendre Polynomials, $P_l^m(x)$')

plt.xlabel(r'$x$')

plt.ylabel(r'$P_l^m(x)$')

plt.legend()

plt.show()

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 The Associated Polynomials on a Polar plot

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import numpy as np

import matplotlib.pyplot as plt

from scipy.special import lpmv

​

ls = [0,1,1,1,2]

ms = [0,-1,0,1,1]

thetas = np.linspace(0,2*np.pi,200)

​

plt.figure(figsize=(8,8))

for i in range(0, len(ls)):

    l=ls[i]

    m=ms[i]

    r = lpmv(m,l,np.cos(thetas))

    plt.polar(thetas, abs(r),label=r'$l=$'+str(l)+r', $m=$'+str(m))

plt.title(r'Associated Legendre Polynomials, $||P_l^m(x)||$')

plt.ylabel(r'$||P_l^m(x)||$')

plt.legend()

plt.show()

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The Associated Polynomials in 3D 

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import numpy as np

import matplotlib.pyplot as plt

from scipy.special import sph_harm

import mpl_toolkits.mplot3d.axes3d as axes3d

import matplotlib.colors as mcolors

​

l=2

m=1

​

thetas = np.linspace(0, np.pi, 20)

phis = np.linspace(0, 2*np.pi, 20)

​

(Theta,Phi)=np.meshgrid(thetas,phis) 

s_harm=sph_harm(m, l, Phi, Theta)

R = abs(s_harm)

X = R * np.sin(Theta) * np.cos(Phi)

Y = R * np.sin(Theta) * np.sin(Phi)

Z = R * np.cos(Theta)

​

cmap = plt.get_cmap('jet')

norm = mcolors.Normalize(vmin=Z.min(), vmax=Z.max())

​

fig = plt.figure(figsize=(8,8))

ax = fig.add_subplot(1,1,1, projection='3d')

ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=plt.get_cmap('jet'),facecolors=cmap(norm(R)),

    linewidth=0, antialiased=False, alpha=0.5)

plt.title(r'Spherical Harmonics, $Y_l^m(\theta,\phi)$'+r', $l=$'+str(l)+r', $m=$'+str(m))

plt.xlabel(r'$x$')

plt.ylabel(r'$y$')

plt.ylabel(r'$z$')

​

plt.show()

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