One-dimensional model of a quantum nonlinear harmonic oscillator

TL;DR

This paper investigates the quantization of a nonlinear harmonic oscillator with position-dependent mass, revealing shape invariance and deriving its energy spectrum, which generalizes the linear harmonic oscillator.

Contribution

It introduces a quantization method for the nonlinear oscillator and demonstrates its shape invariance and spectral properties, extending understanding beyond the linear case.

Findings

The nonlinear oscillator exhibits shape invariance.

Energy spectrum derived via factorization.

Linear harmonic oscillator is a special case as λ→0.

Abstract

In this paper we study the quantization of the nonlinear oscillator introduced by Mathews and Lakshmanan. This system with position-dependent mass allows a natural quantization procedure and is shown to display shape invariance. Its energy spectrum is found by factorization. The linear harmonic oscillator appears as the $\lambda\to 0$ limit of this nonlinear oscillator, whose energy spectrum and eigenfunctions are compared to the linear ones.