Semiclassical quantization of circular billiard in homogeneous magnetic field: Berry-Tabor approach

Gergely Palla,Gábor Vattay,József Cserti

doi:10.1155/s0161171201020129

Abstract

Semiclassical methods are accurate in general in leading order ofħ, since they approximate quantum mechanics via canonical invariants. Often canonically noninvariant terms appear in the Schrödinger equation which are proportional toħ2, therefore a discrepancy between different semiclassical trace formulas in order ofħ2seems to be possible. We derive here the Berry-Tabor formula for a circular billiard in a homogeneous magnetic field. The formula derived for the semiclassical density of states surprisingly coincides with the results of Creagh-Littlejohn theory despite the presence of canonically noninvariant terms.

Highlights

Semiclassical methods are part of an important aspect of quantum chaos, the understanding of the transition from classical to quantum mechanics
The most famous one is Gutzwiller’s trace formula for the level density of classically chaotic systems [9, 10, 11], where the old quantization rules do not apply. This formula relates the density of states to the actions, periods, and stability of classical periodic orbits
Since this method is applicable only when all the involved orbits are isolated in phase space, other methods were developed for systems where periodic orbits form continuous families

Summary

Introduction

Semiclassical methods are part of an important aspect of quantum chaos, the understanding of the transition from classical to quantum mechanics. These methods can be viewed as generalizations of the Bohr-Sommerfeld quantization rules. The most famous one is Gutzwiller’s trace formula for the level density of classically chaotic systems [9, 10, 11], where the old quantization rules do not apply. This formula relates the density of states to the actions, periods, and stability of classical periodic orbits. The Berry-Tabor formula [1, 2] has been developed for integrable systems while the Creagh-Littlejohn theory [4, 5] for systems with continuous symmetries

Objectives

Conclusion