Introduction to Spherical Harmonics

In this video, I go over an introduction to spherical harmonics by giving an overview of how they are obtained from the solution to Laplace's equation in spherical coordinates. The general solution to Laplace's equation (which describes steady-state conditions) is called Solid Spherical Harmonics since it involves the radial term, while excluding it yields the Complex Spherical Harmonics. Finally, defining Real Spherical Harmonics such that the imaginary terms cancel out obtains the common equations for spherical harmonics. And radially distorting spherical harmonics proportional to their values on the sphere obtains the common lobes used in their visualizations and graphs.

#math #sphericalharmonics #calculus #atomicphysics #science

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Timestamps

Spherical harmonics are the solution to Laplace's equation in spherical coordinates – 0:00
Laplace's equation describes steady-state conditions (i.e. in harmony), and solutions are called harmonic functions – 2:00
Separation of variables can solve Laplace's equation to obtain Solid Spherical Harmonics that include the radial term – 4:21
Compacted form of the Associated Legendre Function – 7:47
Angular Laplacian operator excludes the radial terms – 11:28
Solving angular Laplacian obtains the complex Spherical Harmonics with Normalization Constant to ensure a specific double integral is equal to 1 – 13:00
Definition of real spherical harmonics, which cancel out the imaginary terms – 16:55
Normalized real spherical harmonics – 20:16
Full general solution to Laplace's equation is a sum or series over all possible integers n and m – 22:42
Visualizations of Real Spherical Harmonics – 24:22
Can radially distort the sphere proportional to the magnitude of the values on the sphere to get the characteristic lobes – 26:01
Visualization with lobes and colors – 27:56 .