Do equidistant energy levels necessitate a harmonic potential?

TL;DR

This paper examines whether equidistant energy levels necessarily imply a harmonic potential, analyzing experimental data, isospectral potentials, and the limitations of polynomial models in reproducing such spectra.

Contribution

It demonstrates that only the harmonic oscillator potential can produce exactly equidistant energy levels among polynomial potentials and reviews the applicability of various potential models to experimental data.

Findings

Harmonic oscillator uniquely yields exactly equidistant energy levels among polynomial potentials.

Anharmonic and non-polynomial potentials can produce approximately equidistant spectra.

Experimental energy levels alone cannot definitively determine the underlying potential.

Abstract

Experimental results from literature show equidistant energy levels in thin Bi films on surfaces, suggesting a harmonic oscillator description. Yet this conclusion is by no means imperative, especially considering that any measurement only yields energy levels in a finite range and with a nonzero uncertainty. Within this study we review isospectral potentials from the literature and investigate the applicability of the harmonic oscillator hypothesis to recent measurements. First, we describe experimental results from literature by a harmonic oscillator model, obtaining a realistic size and depth of the resulting quantum well. Second, we use the shift-operator approach to calculate anharmonic non-polynomial potentials producing (partly) equidistant spectra. We discuss different potential types and interpret the possible modeling applications. Finally, by applying $n$th order perturbation theory we show that \textbf{exactly} equidistant eigenenergies cannot be achieved by polynomial potentials, except by the harmonic oscillator potential. In summary, we aim to give an overview over which conclusions may be drawn from the experimental determination of energy levels and which may not.