# Spherical Well

### From FSUPhysicsWiki

Let us now consider a spherical well potential, given by

The Schrödinger equations for these two regions are

for and

for

The general solutions are

where and

Let us now consider bound states for the special case, In this case, the centrifugal barrier drops out and the equations become

The solution for this case is

where

Using the boundary condition, we find that The wave functions for thus reduces to

where

For , we know that since, as the wavefunction must go to zero. Therefore, for the region in which

Using the conditions that at the wave functions and their derivatives must be continuous yields the following equations:

and

Dividing the second equation by the first, we obtain

which is just the solution for the odd states in a one-dimensional square well.

This, combined with the fact that

shows that no bound state exists if