Calogero F. Variable Phase Approach to Potential Scattering

New York, London: Academic Press, 1967. — XII, 244 p. — (Mathematics in Science and Engineering Series, Volume 35).The subject of this monograph is nonrelativistic scattering on a spherically symmetrical potential. More specifically, the discussion is focused on the scattering phase shift δ, which, for each value of the angular momentum l, represents all the effect of the potential that is relevant for the quantum mechanical description of elastic scattering. The treatment makes no attempt to go beyond the standards of mathematical rigor that are usually considered satisfactory in physics. In fact, whenever possible the emphasis is on the physical understanding rather than the mathematical formalism. It is nevertheless hoped that the number of imprecise or incorrect statements will be minimal.The fact that a monograph has been written on a subject that is treated in several quantum mechanics textbooks requires some justification. This can be based on the fact that the method employed systematically in this monograph for the investigation of scattering phase shifts (the so-called variable phase method or simply the phase method), although quite old, is almost unknown, while it deserves a much larger audience. In fact, it is elegant, physical, and powerful, because its basic equations are quite simple, have a very direct physical interpretation, and provide a versatile tool for deriving qualitative and quantitative properties of the scattering phase shifts. We believe that this method of discussing scattering phase shifts should be introduced in all elementary quantum mechanics courses that include a treatment of scattering theory.It might be objected that nowadays the exact computation of the scattering phase shifts produced by any potential is a trivial computer job. It is nevertheless important (in fact, probably more so) to acquire a physical feeling for the relation between the potential and the phase shift; this feeling, we believe, is provided by the approach described herein. The importance of such physical intuition can hardly be overestimated; it is certainly very helpful in most problems of current physical interest, including those in high-energy physics where a simple potential model is certainly inadequate to describe the dynamics completely.