Scattering by a Dirac Bubble Potential
For elastic scattering from a spherically symmetrical potential, the asymptotic behavior of the wavefunction can be represented by the Faxen–Holzmark formula[more]
where the first term represents an incoming plane wave in the positive direction () and the second term is an outgoing spherical wave from the scattering center. The energy of a scattered particle is given by
The scattering amplitude determines the angular distribution of the outgoing wave. The differential scattering cross section is given by
For a Dirac bubble potential 
is assumed to be positive (repulsive potential). Negative values of can exhibit bound-state resonances. As shown in [2, 3], the scattering amplitude can be computed from a sum over partial waves in the form
The phase shifts derived in  are given by
where and are spherical Bessel functions. The total scattering cross section is given by
In practice, the summation over partial waves converges very rapidly. We have found it sufficient to truncate the summation at .
This Demonstration shows a pictorial representation of the spherical wavefronts in the Faxen–Holzmark formula, which you can animate. The checkbox superposes a plot of the scattering cross section , shown in red, which changes as , and are varied. You can also display plots of the phase shifts, scattering amplitude and scattering cross section.[less]
 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.
 L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, 2nd ed. (J. B. Sykes and J. S. Bell, trans.), Oxford: Pergamon Press, 1965 Chapter XVII.
 E. Merzbacher, Quantum Mechanics, 3rd ed., New York: John Wiley & Sons, Inc., 1998 Chapter 13.