The electron in a hydrogen atom is described by the Schrödinger equation. The time-independent Schrödinger equation in spherical polar coordinates

can be solved by separation of variables in the form

. The radial and angular components are Laguerre and Legendre functions, thus

and

, respectively. Here,

is the first Bohr radius, and

are the integers in the ranges

(principal quantum number),

(angular momentum quantum number), and

(magnetic quantum number). The probability of finding an electron at a specific location is given by

, where

is the normalization constant, such that

.

This Demonstration shows the three-dimensional distribution of probability

using color when

are specified. Only the real part of

is considered. Putting the proton at the origin, the probability density is displayed within a sphere of radius

. You can select the principal quantum number

in the range 1–3, along with the allowed values of the quantum numbers

and

.