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.