Exploring the vibrant world of spherical harmonics visualization
To conclude my journey on the analysis of the angular momentum, as detailed in my previous posts (here and here), I am thrilled to share my latest exploration into the realm of spherical harmonics visualization. This endeavor aims to extend our mathematical journey from abstract equations to tangible visual representations, crucial for a deeper understanding in various domains of physics and engineering.
Spherical harmonics, characterized by two integers: the degree l and the order m, form an orthonormal basis for the space of square-integrable functions on the sphere. My focus in this blog post is on visualizing the spherical harmonics for l=1, 2, 3, using the Condon-Shortley phase convention. This convention allows us to define the real basis of spherical harmonics Y_{l,m} in a more intuitive manner for visualization and interpretation in numerous applications.
The visualization of spherical harmonics not only aids in our understanding of their mathematical properties but also illustrates their critical role in fields such as quantum mechanics, electromagnetics, and computer graphics. Here is a snippet of my Python code used to visualize these intriguing functions:
from scipy.special import sph_harm
import numpy as np
import matplotlib.pyplot as plt
import sys
import argparse
def plot_Y(ax, l, m):
theta = np.linspace(0, np.pi, 100)
phi = np.linspace(0, 2 * np.pi, 100)
theta, phi = np.meshgrid(theta, phi)
Y = sph_harm(abs(m), l, phi, theta)
if m < 0:
Y = np.sqrt(2) * (-1)**m * Y.imag
elif m > 0:
Y = np.sqrt(2) * (-1)**m * Y.real
In the process of this exploration, I utilized matplotlib for rendering and scipy for spherical harmonics computation, enabling a detailed and high-resolution visualization. This approach revealed the mesmerizing symmetries and complexities of spherical harmonics, further illuminating their beauty and significance.
As we journey through each visualization, from l=0 to l=3, we observe a progression in complexity and symmetry. These visualizations serve as a bridge, connecting the theoretical aspects discussed in earlier posts to the practical applications and implications in physics and engineering.
This blog post signifies not just the culmination of my exploration into angular momentum and spherical harmonics but also a bridge to future explorations in physics and mathematics. The interplay between the theoretical and the visual has opened new avenues for understanding complex phenomena, serving as a testament to the power of interdisciplinary approaches in scientific inquiry.
For more insights into this topic, you can find the details here.