Nonrelativistic one-particle problem on $q$-deformed Euclidean space

TL;DR

This paper explores the quantum mechanics of a free particle on a $q$-deformed Euclidean space, deriving solutions, propagators, and expectation values, thus extending classical quantum theory into a noncommutative geometric framework.

Contribution

It provides explicit plane wave solutions, propagators, and expectation value calculations for the Schrödinger equation on a $q$-deformed space, a novel extension of quantum mechanics.

Findings

Plane wave solutions form a complete orthonormal system.

Derived $q$-deformed propagators for nonrelativistic particles.

Analyzed expectation values of position and momentum.

Abstract

We consider time-dependent Schr\"{o}dinger equations for a free nonrelativistic particle on the three-dimensional $q$-deformed Euclidean space. We determine plane wave solutions to these Schr\"{o}dinger equations and show that they form a complete orthonormal system. We derive $q$-deformed expressions for propagators of a nonrelativistic particle. Considerations about expectation values for position or momentum of a nonrelativistic particle conclude our studies.