Accuracy of the Semi--Classical Approximation: the Pullen Edmonds Hamiltonian

TL;DR

This paper evaluates the numerical accuracy of the semiclassical approximation for the Pullen-Edmonds Hamiltonian, revealing its limitations and comparing it to quantum perturbation theory across different energy levels.

Contribution

It provides a perturbative interpretation of the accuracy decline of the semiclassical approximation at higher quantum numbers for a non-integrable system.

Findings

Semiclassical approximation accuracy decreases with increasing principal quantum number.

On average, semiclassical and quantum perturbation theories yield similar accuracy.

The study offers insights into the limitations of semiclassical methods for complex systems.

Abstract

A test on the numerical accuracy of the semiclassical approximation as a function of the principal quantum number has been performed for the Pullen--Edmonds model, a two--dimensional, non--integrable, scaling invariant perturbation of the resonant harmonic oscillator. A perturbative interpretation is obtained of the recently observed phenomenon of the accuracy decrease on the approximation of individual energy levels at the increase of the principal quantum number. Moreover, the accuracy provided by the semiclassical approximation formula is on the average the same as that provided by quantum perturbation theory.