The eigenfunctions in spherical coordinates for the hydrogen atom are
are the solutions to the radial and angular parts of the Schrödinger equation, respectively, and
are the principal, orbital, and magnetic quantum numbers with allowed values
are the spherical harmonics and the radial functions are
-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.