Quantization of Li\'enard's nonlinear harmonic oscillator and its solutions in the framework of supersymmetric quantum mechanics

TL;DR

This paper quantizes a nonlinear Lie9nard oscillator using supersymmetric quantum mechanics, deriving exact solutions for its energy spectra and wave functions, and showing consistency with the standard harmonic oscillator in the no-deformation limit.

Contribution

It introduces a novel quantization approach for Lie9nard oscillators within SUSYQM, providing explicit solutions and exploring the effects of deformation parameters.

Findings

Exact energy spectra and wave functions derived

Limiting case matches standard quantum harmonic oscillator

Quantization method applicable to nonlinear oscillators

Abstract

Li\'enard-type nonlinear one-dimensional oscillator is quantized using van Roos symmetric ordering recipe for the kinetic-like part of the new derived Hamiltonian. The corresponding Schr\"odinger equation is exactly solved in momuntum space via the approach of supersymmetric quantum mechanics (SUSYQM). The bound-states energy spectra and corresponding wave functions are given explicitly in terms of the ambiguity parameters. The limiting case of no deformation agrees exactly with the eigenenergies and eigenfunctions of the ordinary quantum harmonic oscillator.