In quantum physics, you can apply the spherical Bessel and Neumann functions to a free particle (a particle which is not constrained by any potential). The wave function in spherical coordinates takes this form:

and

gives you the spherical harmonics. The problem is now to solve for the radial part, R* _{nl}*(

*r*). Here's the radial equation:

For a free particle, V(*r*) = 0, so the radial equation becomes

The way you usually handle this equation is to substitute

and because you have a version of the same equation for each *n* index it is convenient to simply remove it, so that R* _{nl }*(

*r*) becomes

This substitution means that

becomes the following:

The radial part of the equation looks tough, but the solutions turn out to be well-known — this equation is called the spherical Bessel equation, and the solution is a combination of the spherical Bessel functions

and the spherical Neumann functions

where A* _{l}* and B

*are constants. So what are the spherical Bessel functions and the spherical Neumann functions? The spherical Bessel functions are given by*

_{l}Here's what the first few iterations of

look like:

How about the spherical Neumann functions? The spherical Neumann functions are given by

Here are the first few iterations of