Particle in a Sphere *

This is like the particle in a box except now the particle is confined to the inside of a sphere of radius $a$. 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.

$$\bgroup\color{black} j_\ell(ka)=0\egroup$$

There are an infinite number of solutions for each $\ell$. We only need to find the zeros of the Bessel functions. The table below gives the lowest values of $ka=\sqrt{2ma^2E\over\hbar^2}$ which satisfy the boundary condition.

$\ell$	$n=1$	$n=2$	$n=3$
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.