Accuracy of the semi-classical approximation: the Pullen-Edmonds Hamiltonian

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.

WKB to all orders and the accuracy of the semiclassical quantization

“…Recently it has been observed by Prosen and Robnik (1993) and also Graffi, Manfredi and Salasnich (1994) that the leading-order semiclassical approximation fails to predict the individual energy levels within a vanishing fraction of the mean energy level spacing. This result has been shown to be true also for the leading (torus) semiclassical approximation by Salasnich and Robnik (1996).…”

Pd–Cu–Ni Ternary Alloy Membrane with High Temperature Durability for Hydrogen Separation and Purification

“…Recently it was observed (Prosen and Robnik 1993, Graffi et al 1994, Robnik and Salasnich 1997a the following this work will be referred to as I) that the torus quantization generally fails to predict the individual energy levels (and the eigenstates) within a vanishing fraction of the mean-energy level spacing. This conclusion is believed to be correct for general systems, including the chaotic ones.…”

On the convergence of the WKB series for the angular momentum operator

“…Recently it was observed (Prosen and Robnik 1993, Graffi, Manfredi and Salasnich 1994, Robnik and Salasnich 1997a the following this work will be referred to as I) that the torus quantization generally fails to predict the individual energy levels (and the eigenstates) within a vanishing fraction of the mean-energy level spacing. This conclusion is believed to be correct for general systems, including the chaotic ones.…”