Schrödinger Equation for a Dirac Bubble Potential
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 .[more]
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.[less]
Snapshot 1: radial function of a continuum state
Snapshot 2: radial function for a bound state
Snapshot 3: contour plot of a continuum
 S. M. Blinder, "Schrödinger Equation for a Dirac Bubble Potential," Chemical Physics Letters, 64(3), 1979 pp. 485–486. doi:10.1016/0009-2614(79)80227-4.