The Schrödinger equation has been solved in closed form for about 20 quantum-mechanical problems. This Demonstration describes one such example published some time ago. A particle moves in a potential that is zero everywhere except on a spherical bubble of radius

, drawn as a red circle in the contour plots. This result has been applied to model the buckminsterfullerene molecule

and also to approximate the interatomic potential in the helium van der Waals dimer

.

The relevant Schrödinger equation is given by

, in units with

,

and

in bohrs, and

in hartrees. For

, the equation has separable continuum solutions

, where the

are spherical harmonics. The radial function has the form

for

and

for

. Here

and

are spherical Bessel functions and the

are phase shifts. For each value of

, a single bound state will exist, provided that

. If no bound state exists, the plot will remain blank. The bound-state radial function is

, where

and

are the greater and lesser of

and

, and

is a Hankel function. The energy is given by

, with

determined by the transcendental equation

. Both the bound and continuum wavefunctions are continuous at

but have discontinuous first derivatives. The second derivative produces a deltafunction.

This Demonstration shows plots of the radial functions

and a cross section of the density plots of

for

. The wavefunction is positive in the blue regions and negative in the white regions. Be cautioned that the density plots might take some time to complete.