Quantum Mechanics on the h-deformed Quantum Plane

TL;DR

This paper develops a covariant deformed Heisenberg algebra and Laplace-Beltrami operator on the h-deformed quantum plane, solving Schrödinger equations and analyzing how deformation affects quantum particle behavior and bound states.

Contribution

It introduces a new covariant deformed algebra and explicitly solves Schrödinger equations on the h-deformed quantum plane, revealing deformation-dependent spectral properties.

Findings

Bound state energies depend explicitly on the deformation parameter h.

Bound states can persist even when they disappear on the Poincaré half-plane.

Quantum particle behavior approaches that on the Poincaré half-plane in the commutative limit.

Abstract

We find the covariant deformed Heisenberg algebra and the Laplace-Beltrami operator on the extended $h$-deformed quantum plane and solve the Schr\"odinger equations explicitly for some physical systems on the quantum plane. In the commutative limit the behaviour of a quantum particle on the quantum plane becomes that of the quantum particle on the Poincar\'e half-plane, a surface of constant negative Gaussian curvature. We show the bound state energy spectra for particles under specific potentials depend explicitly on the deformation parameter $h$. Moreover, it is shown that bound states can survive on the quantum plane in a limiting case where bound states on the Poincar\'e half-plane disappear.