Conservation laws for a $q$-deformed nonrelativistic particle

TL;DR

This paper develops $q$-deformed versions of Green's theorem and derives continuity equations for probability, energy, and momentum densities of a $q$-deformed nonrelativistic particle, extending classical conservation laws into the quantum group framework.

Contribution

It introduces $q$-deformed Green's theorem and derives corresponding conservation laws for a $q$-deformed Schrödinger equation, advancing the mathematical foundation of quantum groups.

Findings

Derived $q$-versions of Green's theorem.

Established continuity equations for $q$-deformed densities.

Extended classical conservation laws to $q$-deformed quantum systems.

Abstract

We derive $q$-versions of Green's theorem from the Leibniz rules of partial derivatives for the $q$-deformed Euclidean space. Using these results and the Schr\"{o}dinger equations for a $q$-deformed nonrelativistic particle, we derive continuity equations for the probability density, the energy density, and the momentum density of a $q$-deformed nonrelativistic particle.