A position-dependent mass harmonic oscillator and deformed space

TL;DR

This paper explores a deformed quantum space framework where a position-dependent mass harmonic oscillator is transformed into a constant mass system in a deformed space, analyzing classical and quantum dynamics with a deformation parameter.

Contribution

It introduces a canonical transformation linking position-dependent mass systems to constant mass systems in deformed space, providing new insights into $q$-deformed oscillators and their classical and quantum behaviors.

Findings

Classical trajectories depend on the deformation parameter.

Quantum analysis via WKB shows correspondence with classical results.

The transformation maps harmonic to Morse potentials in deformed space.

Abstract

We consider canonically conjugated generalized space and linear momentum operators $\hat{x}_q$ and $ \hat{p}_q$ in quantum mechanics, associated to a generalized translation operator which produces infinitesimal deformed displacements controlled by a deformation parameter $q$. A canonical transformation $(\hat{x}, \hat{p}) \rightarrow (\hat{x}_q, \hat{p}_q)$ leads the Hamiltonian of a position-dependent mass particle in usual space to another Hamiltonian of a particle with constant mass in a conservative force field of the deformed space. The equation of motion for the classical phase space $(x, p)$ may be expressed in terms of the deformed (dual) $q$-derivative. We revisit the problem of a $q$-deformed oscillator in both classical and quantum formalisms. Particularly, this canonical transformation leads a particle with position-dependent mass in a harmonic potential to a particle with constant mass in a Morse potential. The trajectories in phase spaces $(x,p)$ and $(x_q, p_q)$ are analyzed for different values of the deformation parameter. Lastly, we compare the results of the problem in classical and quantum formalisms through the principle of correspondence and the WKB approximation.