Scattering of a particle on the $q$-deformed Euclidean space

TL;DR

This paper develops a formalism for particle scattering in $q$-deformed Euclidean space, introducing $q$-versions of key quantum mechanics equations and analyzing their properties.

Contribution

It introduces $q$-deformed versions of the Lippmann-Schwinger equation, Born series, and S-matrix, extending scattering theory to $q$-deformed Euclidean space.

Findings

Derived $q$-versions of scattering equations.

Established unitarity of the $q$-S-matrix.

Discussed $q$-deformed interaction picture and perturbation theory.

Abstract

We develop a formalism for the scattering of a particle on the $q$-deformed Euclidean space. We write down $q$-versions of the Lippmann-Schwinger equation. Their iterative solutions for a weak scattering potential lead us to $q$-versions of the Born series. With the expressions for the wave functions of the scattered particle, we can write down S-matrix elements. We show that these S-matrix elements satisfy unitarity conditions. Considerations about the interaction picture for a quantum system in the $q$-deformed Euclidean space and a discussion of a $q$-version of time-dependent perturbation theory conclude our studies.