Aspects of Quantum Chaos

Abstract

This chapter discusses two important ways of defining quantum chaoticity. One access to characterize dynamical quantum systems is offered by the powerful approach of semiclassics. Here we compare the properties of the quantum system with its classical analogue, and we combine classical intuition with quantum evolution. The second approach starts from the very heart of quantum mechanics, from the quantum spectrum and its properties. Classical and quantum localization mechanisms are presented, again originating either from the classical dynamics of the corresponding problem and semiclassical explanations, or from the quantum spectra and the superposition principle. The essential ideas of the theory of random matrices are introduced. This second way of characterizing a quantum system by its spectrum is reconciled with the first approach by the conjectures of Berry and Tabor and Bohigas, Giannoni, and Schmit, respectively.