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Journal of Mathematical Physics

Probabilistic Interpretation of the Classical Scattering Cross Section

Donald G. Babbitt

Published Jan 1, 1971• 3 citations
Publisher

Abstract

The probabilistic interpretation of the classical scattering cross section is discussed in a mathematically rigorous framework. In particular, extensive use is made of the notion of a Poisson process based on a plane and on a sphere. The case of Rutherford scattering is given as a detailed illustration.
DOI
10.1063/1.1665485

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Quantum Rutherford scattering and scattering of classical waves off Coulomb-like potentials have similar formal structures and can be studied using the same mathematical techniques. In both contexts, the long-range nature of the interaction leads to a divergent total cross-section, which has been interpreted and regularized in various ways in the past literature. We review in detail the origin of this divergence, in both real and multipole spaces, and show that it arises from incorrectly using approximations out of their domain of validity. We also stress that although classical and quantum Rutherford scattering share the same formalism, the natures of the associated physical observables differ. We comment on the role of interference: while interference can be safely neglected in a quantum context (due to the fact that the observable quantity is a flux, and the incoming flux is collimated), in a classical context one expects to see a superposition of transmitted and scattered waves in a broad region downstream of the target, hence a cross-section is not connected to any physically observable quantity.
This is a review article that investigates the three approaches to Rutherford Scattering Theory - Classical Mechanics, Non-Relativistic Quantum Mechanics, and Quantum Field Theory - and their distinctions. In the first sections, the mathematical models are demonstrated and a qualitative analysis of the main differences in each interpretation of the phenomenon is performed. Next, numerical simulations show the effects of dealing with relativistic or non-relativistic limits. The results make it clear that no theory is essentially wrong, but the newer ones are just more complete and consider more relevant parameters.
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A unified description of classical and quantal scattering is presented. This description is unified both in the mathematical language and in the conceptual pictures that are used. Statistical states and observables, represented in phase space, are emphasized while both the time dependent trajectory and stationary state beam pictures are used. Semiclassical-type approximation schemes to the generalized differential cross section are presented for the single and double potential cases. Connection is made with the Born and distorted wave Born approximations, for both quantum and classical mechanics. A similar semiclassical-type approximation scheme for the double potential internal state kinetic cross section is presented. Connection is also made with the 'constant acceleration approximation' of Oppenheim and Bloom. Zwanzig's projection operator method is used to express the kinetic cross sections in terms of memory effects for the interacting observables.