Change of Basis Matrix: Cartesian to Spherical Laplacian

Question

I was looking at how a change of basis matrix, $[P_{\beta\leftarrow\alpha}]$, is made. While this is a bit more advanced that than what was taught at the course, I wonder what would be the change of basis matrix of the Laplacian from Cartesian coordinates to spherical ones.

I use the Laplacian in spherical coordinates a lot. It's as common as the plague for the 3-dimensional Schrodinger equation, but I see it kind of depressing how I don't know the underlying beauty of such a thing.

Answer 1

Does Sect. 1.4 on this page answer your question?
http://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Polar_Coords/Polar_Coords.htm (The site's author is most likely Prof. A.F. Bower.)