Evaluation of cross section of elastic scattering for non-relativistic and relativistic particles by means of fundamental scattering formulas

TL;DR

This paper applies the Lippmann-Schwinger scattering equation uniformly to both non-relativistic and relativistic particles to evaluate elastic scattering cross sections, providing a clearer physical interpretation and explicit approximations.

Contribution

It introduces a unified approach using the Lippmann-Schwinger equation for both low-momentum and relativistic particles, enhancing clarity and consistency in scattering calculations.

Findings

Lippmann-Schwinger equation applied to relativistic particles

Cross sections evaluated to first-order Born approximation

Provides explicit Green's functions for relativistic particles

Abstract

In evaluating differential cross section of elastic scattering, different theories were applied to low-momentum and relativistic particles. For low-momentum motion, Lippmann-Schwinger scattering equation was applied, called fundamental formula; while for relativistic particles, a general scattering theory was used which calculates S matrix. In this paper, Lippmann-Schwinger equation is applied uniformly to both low-momentum and relativistic particles. The cross sections are valuated to the first order of Born approximation. One-body time-independent Green's functions for relativistic free particles are given. Compared to the general scattering theory, the fundamental theory has a clearer physical picture and the approximations made are more explicit.