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Journal of Mathematical Physics

Cross sections in quantum mechanics

Enrique Campesino-Romeo, John R. Taylor

Published Jun 1, 1975• 3 citations
Publisher

Abstract

The definition of scattering cross sections requires an averaging over wavepackets with random impact parameters ρ; this leads to an integral of the scattering probability over all ρ in a plane perpendicular to the incident beam. We show that, for scattering off a potential which is O (1/rβ) as r→∞, the scattering probability is O (1/ρ2β−4) as ρ→∞. Thus for any β ≳ 3, the integral over impact parameters is well-defined and convergent.

DOI
10.1063/1.522672

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Progress of Theoretical Physics•Mar 1, 1979•0 citations
The definition of scattering cross sections requires an averaging over wavepackets with random impact parameters ρ; this leads to an integral of the scattering probability over all ρ in a plane perpendicular to the incident beam. We show that, for scattering off a potential which is O (1/rβ) as r→∞, the scattering probability is O (1/ρ2β−4) as ρ→∞. Thus for any β ≳ 3, the integral over impact parameters is well-defined and convergent.
Stationary states are unavoidable from the practical viewpoint, since all the computations of observables, such as cross sections and other experimentally measurable physical quantities, are carried out within timeindependent scattering theory. In a scattering experiment, one attempts to determine the momentum of the impact projectile as precisely as possible. However, in the standard formulation of scattering theory, one tries to reach the same ideal goal by identifying the initial state in the remote past with the plane wave. The problem of the asymptotic convergence of scattering states cannot be formulated in the same way for both proper and improper state vectors. The optical theorem is a convenient way of computing the total cross section Q, when one knows the value of the imaginary part of the scattering amplitude in the forward direction. The scattered particles are moved away from the incident beam for a certain amount which is proportional to the value of the total cross section Q.
We consider scattering by screened Coulomb and screened short-range potentials. We prove, in Born approximation, that the cross section for a sharply cutoff Coulomb potential does not converge to the Rutherford cross section as the screening radius ρ→∞. On the other hand, for certain ’’smooth’’ screening functions we show that the screened Coulomb cross section does approach the Rutherford cross section as ρ→∞. In the case of short-range potentials, we prove (exactly, not just in Born approximation) that the screened cross section approaches the unscreened cross section as ρ→∞, whether the screening function is smooth or not.
The phase-space formulation of quantum scattering is discussed and the asymptotic quantities describing potential scattering in one and three dimensions (transmittances, cross sections) are analyzed. It is found that under certain conditions the transmittance in time-dependent scattering only depends on the momentum distribution of the incident packet, being independent of its particular shape in coordinate space. For three-dimensional scattering, generalized cross sections are defined for momentum coherences and a physical interpretation of the generalized optical theorem is proposed.
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.