Quantum chaology

In Section 11.1 we discuss recent advances in quantum chaology, i.e. the semiclassical basis for the analysis of atomic and molecular spectra in the classically chaotic regime. In Section 11.2 we discuss some recent results in type II quantum chaos within the framework of the dynamic Born-Oppenheimer approximation. Recent experimental and theoretical results of the hydrogen atom in strong microwave and magnetic fields are presented in Sections 11.3 and 11.4, respectively. We conclude this chapter with a brief review of the current status of research on chaos in the helium atom. 

Quantized chaos, or quantum chaology (see Section 4.1), is about understanding the quantum spectra and wave functions of classically chaotic systems. The semiclassical method is one of the sharpest tools of quantum chaology. As discussed in Section 4.1.3 the central problem of computing the semiclassical spectrum of a classically chaotic system was solved by Gutzwiller more than 20 years ago. His trace formula (4.1.72) is the basis for all semiclassical work on the quantization of chaotic systems. 

There are currently two main streams of quantum chaology the fundamental and the applied. Fundamental research in quantum chaology addresses the question of the analytical structure and the convergence 

On the applied side of quantum chaology we find serious efforts to forge the semiclassical method into a handy tool for easy use in connection with arbitrary classically chaotic systems. Quite frankly, the current status of semiclassical methods is such that they are immensely helpful in the interpretation of quantum spectra and wave functions, but are only of limited power when it comes to accurately predicting the quantum spectrum of a classically chaotic system. In this case numerical methods geared toward a direct numerical solution of the Schrodinger equation are easier to handle, more transparent, more accurate and cheaper than any known semiclassical method. It should be the declared aim of applied semiclassics to provide methods as handy and universal as the currently employed numerical schemes to solve the spectral problem of classically chaotic quantum systems. 

Another topic of current interest concerns quantum chaology in higher dimensions. The semiclassical method is expected to be accmate only to order independent of dimension. The level density of a -dimensional system scales like which imphes a level spacing h. Therefore, if we use conventional semiclassical methods to predict energy levels in higher dimensions, we have to be prepared for a relative error in the level spacing that scales according to Thus, in the hmit 0, the error 

---

Keywords Quantum chaology photoelectric effect decoherence mesoscopic systems. 

Thus increasing the temperature leads to a smooth transition from a Poissonian to a Gaussian form in the level spacing distribution. This fact makes finite-temperature treatment more interesting from the viewpoint of quantum chaology. 

Heller, E. J. (1991), Wavepacket Dynamics and Quantum Chaology, in Chaos and Quantum Physics, M.-J. Giannoni et al. (Eds.) Elsevier. 

The plan of Chapter 5 is the following. In order to get a feeUng for the dynamics of the kicked molecule, we approximate it by a one-dimensional schematic model by restricting its motion to rotation in the x, z) plane and ignoring motion of the centre of mass. In this approximation the kicked molecule becomes the kicked rotor, probably the most widely studied model in quantum chaology. This model was introduced by Casati et al. in 1979. The classical mechanics of the kicked rotor is discussed in Section 5.1. Section 5.2 presents Chirikov s overlap criterion, which can be applied generally to estimate analytically the critical control parameter necessary for the onset of chaos. We use it here to estimate the onset of chaos in the kicked rotor model. The quantum mechanics of the kicked rotor is discussed in Section 5.3. In Section 5.4 we show that the results obtained for the quantum kicked rotor model are of immediate... 

Berry, M.V. (1989). Quantum chaology, not quantum chaos, Physica Scripta 40, 335-336. 

Keating, J. (1993). The Riemann zeta function and quantum chaology, in Quantum Chaos, eds. G. Casati, I. Guarneri and U. Smilansky (North-Holland, Amsterdam). 

Excellent reviews exist addressing the topic of chaos in atomic physics. The most comprehensive attempt at a review of the entire subject of quantum chaology and applications is a recent collection of reprints and original articles edited by Casati and Chirikov (1995). Its section on atoms in strong fields especially is highly relevant for the topic of this book. Additional material on chaos and irregularity in atomic physics can be found, e.g., in a collection of articles edited by Gay (1992). 

Thus, one is interested in a rather special kind of statistics, viz. the statistics of a dense population of interacting levels. This is the fundamental distinction between chaos and complexity there may arise situations in which levels do not necessarily all interact (they might have different quantum numbers) but are simply present in large numbers, so that their analysis is not possible in practice but could be performed in principle. These are called unresolved transition arrays (UTAs). One can develop [526] a theory of UTAs which yields general theorems about them as a whole. Such theories are a statistical approach to the interpretation of spectra, but are not related to the problem of quantum chaology. 

Although Garton and Tomkins discovered -mixing, n-mixing and the quasi-Landau resonances in the spectrum of Ba, from the standpoint of quantum chaology the asymptotically Coulombic nature of the atomic... 

---