Solid Spherical Harmonics – Solutions to Laplace's Equation in Spherical Coordinates

In this video, I go over the derivation of the solid spherical harmonics, which are solutions to the Laplace equation in spherical harmonics. They are referred to as "solid" because they include the radial term as well. I use separation of variables to define a function that is a multiple of 3 separated functions, and then selected separation constants to obtain a periodic, repeating solution. I solve the function corresponding to the azimuthal angle ɸ via the derivative of an exponential function. The function containing the radial r term is the Euler-Cauchy equation, and I solve it via substitution. The middle function is the associated Legendre equation, whose solution is beyond the scope of this video, so I just plugged in the corresponding function. Combining all of these obtains our solid spherical harmonics!

#math #sphericalharmonics #calculus #quantumphysics #laplace