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