For elastic scattering from a spherically symmetrical potential, the asymptotic behavior of the wavefunction can be represented by the Faxen–Holzmark formula

,

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 [1]

,

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 [1] 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.

[1] 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.

[2] 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.

[3] E. Merzbacher, Quantum Mechanics, 3rd ed., New York: John Wiley & Sons, Inc., 1998 Chapter 13.