Rutherford scattering of quantum and classical fields

TL;DR

This paper compares quantum and classical Rutherford scattering, clarifies the origin of divergences in total cross-section calculations, and highlights differences in physical observables and interference effects between the two contexts.

Contribution

It provides a detailed review of the divergence origins in Rutherford scattering and emphasizes the differences in physical observables between classical and quantum cases.

Findings

Divergence in total cross-section arises from invalid approximations.

Classical and quantum Rutherford scattering share formalism but differ in physical observables.

Interference effects are negligible in quantum but significant in classical scattering.

Abstract

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.