The quantized Baker's transformation
type: Item
instance of: scholarly article
title: The quantized Baker's transformation
zbMATH Open document ID: 0664.58045
DOI: 10.1016/0003-4916(89)90259-5
author: N. L. Balazs
author: André Voros
published in: Annals of Physics
publication date: 1989-01-01T00:00:00Z
review text: A quantum analogue of the Baker's transformation is constructed using a specially developed quantization procedure. We obtain a unitary operator acting on an N-dimensional Hilbert space, with N finite (and even), that has similar properties to the classical Baker's map, and reduces to it in the classical limit, which corresponds here to \(N\to \infty\). The operator can be described as a very simple, fully explicit \(N\times N\) matrix. Numerical investigations confirm that this model has non-trivial features which ought to represent quantal manifestations of classical chaoticity. The quasi-energy spectrum is given by irrational eigenangles, leading to no recurrences. Most eigenfunctions look irregular, but some exhibit puzzling regular features, such as peaks at coordinate values belonging to periodic orbits of the classical Baker's map. We compare the quantal and classical time-evolutions, as applied to initially coherent quasi-classical states: the evolving states stay in close agreement for short times but seem to lose all relationship to each other beyond a critical time of the order of \(\log_ 2N\sim -\log \hslash\).
Mathematics Subject Classification ID: 58Z05
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4086608
zbMATH Keywords: Baker's transformation
zbMATH Keywords: quantization
zbMATH Keywords: quasi-energy spectrum
zbMATH Keywords: eigenfunctions
MaRDI profile type: Publication
full work available at URL: 0003-4916(89)90259-5
OpenAlex ID: W2003964615
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description: scientific article; zbMATH DE number 4086608