Spherical Harmonics – Solutions to the Angular Laplacian

mes (67)in DTube • 2 days ago

In this video, I go over the actual spherical harmonics equations, which exclude the radial terms from the solid spherical harmonics I had derived earlier. The spherical harmonics can be combined into one function that contains all of the angular terms, thus essentially treating the sphere's radius as a constant. I also show how the spherical harmonics can be considered the solution of a modified Laplace equation, and hence a modified Laplace operator termed the angular Laplacian.

#math #sphericalharmonics #calculus #quantumphysics #laplace

Timestamps:

Spherical harmonics excludes the radial term and only has the angular terms – 0:00
Combine the radial terms into a function Y – 1:02
Y is considered the solution to the angular Laplacian – 2:44
Angular Laplacian excludes the radial terms – 4:33
Factor out the Y term from the radial term – 6:47
Choose separation constants just as with solid spherical harmonics – 7:28
Obtain the typical equation whose solution Y is the spherical harmonics – 8:56
Equation for Y, defined earlier – 10:13

Notes and playlists: