Wave packets in quantum theory of collisions

TL;DR

This paper addresses two key issues in quantum collision theory by employing wave packets, resolving interference problems in stationary approaches and mathematical difficulties in nonstationary methods.

Contribution

It introduces wave packets to effectively resolve interference and mathematical issues in quantum collision calculations, improving the theoretical framework.

Findings

Wave packets eliminate interference between incident and scattered waves.

Wave packets regularize the delta-squared problem in S-matrix calculations.

The approach enhances the mathematical consistency of collision theory.

Abstract

Two methodological troubles of the quantum theory of collisions are considered. The first is the undesirable interference of the incident and scattered waves in the stationary approach to scattering. The second concerns the nonstationary approach to the theory of collisions of the type $a+b\to c+d$. In order to calculate the cross section one uses the matrix element $<cd|S|ab>$ of the $S$-matrix. The element is proportional to $\delta$-function expressing the energy conservation. The corresponding probability $|< cd|S|ab>|^2$ contains $\delta^2$ which is mathematically senseless. The known regular way to overcome the difficulty seems to be unsatisfactory. In this paper, both the troubles are resolved using wave packets of incident particles.