Hydrogen Atom Radial Functions

The eigenfunctions in spherical coordinates for the hydrogen atom are , where and are the solutions to the radial and angular parts of the Schrödinger equation, respectively, and , , and are the principal, orbital, and magnetic quantum numbers with allowed values , and . The are the spherical harmonics and the radial functions are , where is the -order associated Laguerre polynomial and is the Bohr radius. The left graphic shows the radial probability density and the expectation value , and the right graphic shows the radial function.