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European Journal of Physics

Simple quantum calculation of the Rutherford cross section

N. Gauthier

Published Mar 1, 2000• 5 citations
Publisher

Abstract

An elementary quantum-mechanical calculation of the Rutherford scattering cross section is presented. The approach can be followed by a student who is familiar with the Fermi rule for first-order transitions and it does not require knowledge of more sophisticated techniques of quantum collision theory. A close parallel is maintained between the present quantum definition of the scattering cross section and its more familiar classical counterpart.

DOI
10.1088/0143-0807/21/2/403

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An elementary quantum-mechanical calculation of the Rutherford scattering cross section is presented. The approach can be followed by a student who is familiar with the Fermi rule for first-order transitions and it does not require knowledge of more sophisticated techniques of quantum collision theory. A close parallel is maintained between the present quantum definition of the scattering cross section and its more familiar classical counterpart.
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.
THE classical theory of a point charge and the quantum theory of wave packets contain in their usual form well-known divergences. Dirae1 has elaborated a classical particle theory which avoids these difficulties, but his results cannot as yet be applied to quantum theory. On the other hand, Heisenberg1 has recently proposed a new formalism which permits the calculation of collision cross-sections in quantum theory without being disturbed by diverging terms. However, no connecting link such as the correspondence principle has been given in order to apply this quantum formalism to a given problem such as Rutherford scattering, Compton effect or radiation damping.
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.
In the scattering of three free incident particles, a finite cross section for the double scattering has to be defined differently from the processes other than the double scattering. That is, we want to maintain that, in order to obtain a definite quantity being connected to the experimental observation, a particular definition of the cross section has to be required for the double scattering owing to a particular time dependence of its scattering rate. Our definition differs from the ones given recently by others in a form depending on some macroscopic quantities.1), 2) In the previous paper (refered as I in the following)3), we have attained to a finite cross section by working on the basis of the wave packet formalism, while we want to give here a clear-cut derivation in a simple and familear form.