Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Introduction
- Part One Classical chaos and quantum localization
- Part Two Atoms in strong fields
- Part Three Semiclassical approximations
- Semiclassical theory of spectral rigidity
- Semiclassical structure of trace formulas
- ħ-Expansion for quantum trace formulas
- Pinball scattering
- Logarithm breaking time in quantum chaos
- Semiclassical propagation: how long can it last?
- The quantized Baker's transformation
- Classical structures in the quantized baker transformation
- Quantum nodal points as fingerprints of classical chaos
- Chaology of action billiards
- Part Four Level statistics and random matrix theory
- Index
Classical structures in the quantized baker transformation
Published online by Cambridge University Press: 07 May 2010
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Introduction
- Part One Classical chaos and quantum localization
- Part Two Atoms in strong fields
- Part Three Semiclassical approximations
- Semiclassical theory of spectral rigidity
- Semiclassical structure of trace formulas
- ħ-Expansion for quantum trace formulas
- Pinball scattering
- Logarithm breaking time in quantum chaos
- Semiclassical propagation: how long can it last?
- The quantized Baker's transformation
- Classical structures in the quantized baker transformation
- Quantum nodal points as fingerprints of classical chaos
- Chaology of action billiards
- Part Four Level statistics and random matrix theory
- Index
Summary
We study the role of periodic trajectories and other classical structures on single eigenfunctions of the quantized version of the baker's transformation. Due to the simplicity of both the classical and the quantum description a very detailed comparison is possible, which is made in phase space by means of a special positive definite representation adapted to the discreteness of the map. A slight but essential modification of the original version described by Balasz and Voros (Ann. Phys. 190 (1989), 1) restores the classical phase space symmetry. In particular, we are able to observe how the whole hyperbolic neighborhood of the fixed points appears in the eigenfunctions. New scarring mechanisms related to the homoclinic and heteroclinic trajectories are observed and discussed. © 1990 Academic Press, Inc.
INTRODUCTION
In this paper we provide a detailed analysis of the classical structures that we are able to recognize in single eigenfunctions of a strongly chaotic system, the quantized version of the baker transformation [1], thus contributing to an answer to the question “what are single eigenfunctions of classically chaotic systems made of?” If we were to ask this question for an integrable system the answer would be immediate: the wave functions peak on the invariant tori quantized by discrete values of the actions. Thus each wave function shows the imprint of one torus which becomes more and more sharply defined as the classical limit is approached.
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- Quantum ChaosBetween Order and Disorder, pp. 483 - 506Publisher: Cambridge University PressPrint publication year: 1995