Details

For simplicity, set (atomic units) and represent the energy by . Clearly the Schrödinger equation is separable in spherical polar coordinates and we can write

,

where is a spherical harmonic. The radial functions then obey the equations:

for

and

for .

The solutions are spherical Bessel and Hankel functions:

for

and

for ,

where

and

.

The eigenfunctions are determined by the continuity conditions on the wavefunctions and their derivative at : and . This can be simplified to

.

For , this leads to a transcendental equation

.

For , the corresponding relation is

.

Solutions for real correspond to one of the finite number of bound states (with ).

References

[1] J. Branson. "Bound States in a Spherical Potential Well." (Jul 31, 2019) quantummechanics.ucsd.edu/ph130a/130_notes/node227.html.

[2] "Finite Spherical Square Well Bound States." (Jul 31, 2019) www.physics.csbsju.edu/QM/square.16.html.