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CRC Press eBooks

Concluding remarks

M. Brack, Rajat K. Bhaduri

Published Mar 5, 2018• 0 citations

Abstract

The M. C. Gutzwiller trace formula has played a prominent role in the study of quantum systems whose classical analogues are chaotic. Although of measure zero in a sea of chaos in the classical phase space, the periodic orbits still open a classical window to view the quantum behavior of such systems. One of the earliest applications of the trace formula in this connection was made by Strutinsky and collaborators in their studies of deformed atomic nuclei. The same physics of the dominance of short periodic orbits in finite-sized systems makes the trace formula a powerful tool in the study of mesoscopic devices. Fascinating quantal behavior in the propagation of wave packets, like the phenomenon of recurrence, is explained by a semiclassical analysis. Diffractive orbits are specially important for grazing angles of encounter and, of course, play a leading role in scattering in the forward direction.

DOI: 10.1201/9780429502828-9

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The M. C. Gutzwiller trace formula has played a prominent role in the study of quantum systems whose classical analogues are chaotic. Although of measure zero in a sea of chaos in the classical phase space, the periodic orbits still open a classical window to view the quantum behavior of such systems. One of the earliest applications of the trace formula in this connection was made by Strutinsky and collaborators in their studies of deformed atomic nuclei. The same physics of the dominance of short periodic orbits in finite-sized systems makes the trace formula a powerful tool in the study of mesoscopic devices. Fascinating quantal behavior in the propagation of wave packets, like the phenomenon of recurrence, is explained by a semiclassical analysis. Diffractive orbits are specially important for grazing angles of encounter and, of course, play a leading role in scattering in the forward direction.

Gutzwiller's trace formula has a central place in quantum chaos because it provides semiclassical approximations for quantum energy levels in classically chaotic systems by linking them to classical periodic orbits. In this didactic article, we discuss a derivation of the trace formula starting from the Feynman path integral. We then describe how the trace formula is used to explain universal features in the distribution of the quantum energy levels that are described by random matrix theory, and we give an overview of related work.

We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate-space formulation. Using a coherent-state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum-mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

Abstract We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate‐space formulation. Using a coherent‐state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum‐mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

Mesoscopic devices, with system sizes in the range of several to several dozens wavelengths, represent paradigmatic model systems for the observation of quantum chaotic behaviour based on semiclassical concepts. Those electronic and photonic billiard cavities are small enough for interference effects not to be ignored. Nonetheless, the classical ray or particle tracing picture can often provide a substantial understanding of the dynamics of the system along the lines of classical-quantum, or ray-wave correspondence. This well-established principle turns out to be particularly useful when applied not only in real space, but by extending it to phase space such that both location and momentum information can contribute to a deeper and more comprehensive understanding of the dynamical behaviour.

This book introduces the quantum mechanics of classically chaotic systems, or quantum chaos for short. The author's philosophy has been to keep the discussion simple and to illustrate theory, wherever possible, with experimental or numerical examples. The microwave billiard experiments, initiated by the author and his group, play a major role in this respect. Topics covered include the various types of billiard experiment, random matrix theory, systems with periodic time dependences, the analogy between the dynamics of a one-dimensional gas with a repulsive interaction and spectral level dynamics, where an external parameter takes the role of time, scattering theory distributions and fluctuation, properties of scattering matrix elements, semiclassical quantum mechanics, periodic orbit theory, and the Gutzwiller trace formula. This book will be of great value to anyone working in quantum chaos.