Toy model of particle scattering theory

TL;DR

This paper introduces a one-dimensional probabilistic toy model for particle scattering, linking scattering probability to hypervolume and solving it via Mellin transformation, revealing symmetry properties.

Contribution

It presents a novel toy model for particle scattering, providing analytical solutions and uncovering symmetry invariance not previously explored.

Findings

Scattering probability equals the hypervolume of an n-dimensional figure.

Solution for the toy model is expressed as a contour integral via Mellin transformation.

The model exhibits a symmetry invariance concerning initial particle positions.

Abstract

The one dimensional probabilistic toy model of particle scattering theory is proposed. The toy model version of scattering probability is proved to be equal to the hypervolume of a n-dimensional figure. The solution for any n-particle toy model is presented as a contour integral, through Mellin trasnformation. The method of solving the contour integral is discussed. A nontrivial symmetry of this toy model, the invariance on initial position of the particles, is observed.