Quantum Harmonic Oscillators with Nonlinear Effective Masses

TL;DR

This paper investigates quantum harmonic oscillators with a nonlinear, density-dependent effective mass, revealing continuous spectra and novel peakon-like solutions, thus opening new avenues in quantum physics modeling.

Contribution

It introduces a model of quantum harmonic oscillators with nonlinear effective masses, providing analytical solutions and discovering new peakon-like states.

Findings

Continuous energy spectra due to nonlinear effective mass

Agreement between analytical and numerical solutions

Discovery of peakon-like solutions without linear counterparts

Abstract

We study the eigen-energy and eigen-function of a quantum particle acquiring the probability density-dependent effective mass (DDEM) in harmonic oscillators. Instead of discrete eigen-energies, continuous energy spectra are revealed due to the introduction of a nonlinear effective mass. Analytically, we map this problem into an infinite discrete dynamical system and obtain the stationary solutions by perturbation theory, along with the proof on the monotonicity in the perturbed eigen-energies. Numerical results not only give agreement to the asymptotic solutions stemmed from the expansion of Hermite-Gaussian functions, but also unveil a family of peakon-like solutions without linear counterparts. As nonlinear Schr{\"o}dinger wave equation has served as an important model equation in various sub-fields in physics, our proposed generalized quantum harmonic oscillator opens an unexplored area for quantum particles with nonlinear effective masses.