A particle of mass

in an infinite spherical potential well of radius

is described by the Schrödinger equation

. The wavefunction is separable in spherical polar coordinates, such that

, where

is a spherical harmonic,

a spherical Bessel function, and

is a normalization constant. The boundary condition that

at

is fulfilled when

is the

zero of the spherical Bessel function

. The quantized energy levels are then given by

and are

-fold degenerate with

. The conventional code is used to label angular momentum states, with

representing

. Unlike atomic orbitals, the

-values are not limited by

; thus one will encounter states designated

, etc.

This Demonstration shows contour plots on a cross section of the sphere for the lower-energy eigenfunctions with

and

. For

, the eigenfunctions are complex. In all cases, the real parts of

are drawn. The wavefunction is positive in the blue regions and negative in the white regions. You can also view an energy-level diagram, with each dash representing the degenerate set of eigenstates for given

.