Analytic Model for the Energy Spectrum of the Anharmonic Oscillator

TL;DR

This paper derives an analytic energy spectrum for anharmonic oscillators, generalizing previous work on quartic potentials, with results showing about a few percent accuracy and no adjustable parameters.

Contribution

It provides a new analytic expression for the energy spectrum of general anharmonic oscillators, extending previous models and demonstrating high accuracy with minimal computational effort.

Findings

Energy levels are obtained with about a 3-5% relative error.

The model applies to potentials of the form V(x)=ω^2/2 x^2 + g x^{2m}.

The approach is simple, parameter-free, and computationally efficient.

Abstract

In a recent work we have proposed an original analytic expression for the partition function of the quartic oscillator. This partition function, which has a simple and compact form with {\it no adjustable parameters}, reproduces some key mathematical properties of the exact partition function and provides free energies accurate to a few percent over a wide range of temperatures and coupling constants. In this work, we present the derivation of the energy spectrum of this model. We also generalize our previous study limited to the quartic oscillator to the case of a general anharmonic oscillator. Numerical application for a potential of the form $V(x)=\frac{\omega^2}{2} x^2 + g x^{2m}$ show that the energy levels are obtained with a relative error of about a few percent, a precision which we consider to be quite satisfactory given the simplicity of the model, the absence of adjustable parameters, and the negligible computational cost.