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American Journal of Physics

Scattering by spherically symmetric and nonspherically symmetric potentials

A. Ziegler

Published Jun 1, 1989• 3 citations
Publisher

Abstract

The usual presentation of scattering in standard quantum mechanics texts—based on the scattering phase shift δl —is neither very enlightening nor does it lend itself readily to generalization to nonspherically symmetric potentials. The presentation given in this article avoids these drawbacks by being both more general and more transparent.

DOI
10.1119/1.15987

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The usual presentation of scattering in standard quantum mechanics texts—based on the scattering phase shift δl —is neither very enlightening nor does it lend itself readily to generalization to nonspherically symmetric potentials. The presentation given in this article avoids these drawbacks by being both more general and more transparent.
The variable phase approach to potential scattering with regular spherically symmetric potentials satisfying (1), and studied by Calogero in his book5, is revisited, and we show directly that it gives the absolute definition of the phase-shifts, i.e. the one which defines δℓ(k) as a continuous function of k for all k ≥ 0, up to infinity, where δℓ(∞) = 0 is automatically satisfied. This removes the usual ambiguity ±nπ, n integer, attached to the definition of the phase-shifts through the partial wave scattering amplitudes obtained from the Lippmann-Schwinger integral equation, or via the phase of the Jost functions. It is then shown rigorously, and also on several examples, that this definition of the phase-shifts is very general, and applies as well to all potentials which have a strong repulsive singularity at the origin, for instance those which behave like gr−m, g> 0, m ≥ 2, etc. We also give an example of application to the low-energy behaviour of the S-wave scattering amplitude in two dimensions, which leads to an interesting result.
Although the general properties of the phase shift of the three-dimensional spherically symmetrical square well potential are described in most textbooks on quantum mechanics, specific graphical representations of this function are rare. In this paper we present a number of phase shifts, as a function of energy, for the various values of the angular momentum quantum number. We discuss the general features of these curves. In particular, we consider what happens to the phase shifts if one varies the depth of the well, not only for near-zero energy, but for the entire range of energies. We describe how the bound levels of a deep well are related to the virtual levels of the shallow well. We find points of triple coincidence for curves belonging to three successive l values for given well depth. A proof of this phenomenon is given.