This is like the particle in a box except now the
particle is confined to the inside of a sphere of radius
.
Inside the sphere, the solution is a Bessel function.
Outside the sphere, the wavefunction is zero.
The boundary condition is that the wave function go to zero on the sphere.
There are an infinite number of solutions for each
.
We only need to find the zeros of the Bessel functions.
The table below gives the lowest values of
which satisfy the boundary condition.




0 
3.14 
6.28 
9.42 
1 
4.49 
7.73 

2 
5.72 
9.10 

3 
6.99 
10.42 

4 
8.18 


5 
9.32 


We can see both angular and radial excitations.
Jim Branson
20130422