Spherical Box with Hole

A particle is confined to a spherical box of radius R. There is a barrier in the center of the box, which excludes the particle from a radius a. So the particle is confined to the region a < r < R. Assume that the wave function vanishes at both r = a and r = R and derive an expression for the eigenvalues and eigenfunctions of states with angular momentum ℓ = 0.

SOLUTION

In spherical coordinates the eigenfunctions for noninteracting particles of wave vector k are of the form

(1)

where j_ℓ and η­_ℓ are spherical Bessel functions. The constants A and B are determined by the boundary conditions. Since we were only asked for the states with ℓ = 0, we only need j₀(z) = sin z/z and η₀(z) = -cos z/z. We can take a linear combination of these functions, which is a particular choice of the ratio B/A, to make the wave function vanish at r = a.

(2)

This satisfies the boundary condition at r = a. Requiring that this function vanish at r = R gives k(R – a) = nπ, so

(3)

(4)