Exact solution and coherent states of an asymmetric oscillator with position-dependent mass

TL;DR

This paper analyzes an asymmetric oscillator with position-dependent mass, mapping it to a Morse oscillator, and studies its classical and quantum properties, including bound states, scattering states, and coherent state behavior.

Contribution

It introduces a novel approach by incorporating the mass function into both kinetic and potential energies and explores the resulting coherent states and their dynamics.

Findings

Bound trajectories correspond to anharmonic oscillations.

Coherent states exhibit fast localization in phase space.

Uncertainty oscillates with increasing amplitude as deformation grows.

Abstract

We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. {\bf 62}, 092101 (2021)] in the classical and quantum formalisms, by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space $(x, p)$ to a deformed one $(x_\gamma, \Pi_\gamma)$. Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.