Momentum and Position Representations for the q-deformed Euclidean Quantum Space

TL;DR

This paper explores the mathematical structure of q-deformed Euclidean space, demonstrating the completeness and orthonormality of q-deformed eigenfunctions for momentum and position, and analyzing their physical implications.

Contribution

It introduces a framework for analyzing q-deformed eigenfunctions, showing their completeness, orthonormality, and calculating matrix elements within the star product formalism.

Findings

q-deformed eigenfunctions are complete and orthonormal

Matrix elements of momentum and position operators are computed

Expectations and probability densities are analyzed

Abstract

We summarize some basics about mathematical tools of analysis for the q-deformed Euclidean space. We use the new tools to examine q-deformed eigenfunctions of the momentum or position operator within the framework of the star product formalism. We show that these two systems of functions are complete and orthonormal. With the q-deformed momentum or position eigenfunctions, we calculate matrix elements of the momentum or position operator. Considerations about expectation values and probability densities conclude the studies.