On the quantum Fermi accelerator and its relevance to ``quantum chaos''
type: Item
instance of: scholarly article
title: On the quantum Fermi accelerator and its relevance to ``quantum chaos''
zbMATH Open document ID: 0698.35143
DOI: 10.1007/BF00399758
author: N. E. Zubov
published in: Letters in Mathematical Physics
publication date: 1989-01-01T00:00:00Z
review text: For a particle motion bouncing from walls an annoying phenomenon has bathered the experts. While quantum mechanical interpretation predicts stable behavior, the classical mechanics predicts chaotic motion. The author argues that the quantum mechanical model using the Schrödinger equation suppresses stochasticity. The usual technique for solving the problem consists in expanding the solution of the time- dependent Schrödinger equation into eigenmodes of the static equation while ``fixing'' the boundary conditions. Both intuitive arguments and study of the nonlinearity of perturbations indicate that such approach is too restrictive. A new time variable is introduced. It is equal to the conserved energy of the closed system. A quasi-energy operator is defined. It is so-called Floquet-Hamiltonian. The setting is \(L^ 2\), the Schrödinger operator is \(k=-i\partial_ s-\partial^ 2_ x\), where s denotes the physical time, while t denotes the new time scale. After some preliminary results concerning the corresponding one-parameter group the author establishes a relation between the operator exp(-ikt) and the propagator of the original Schrödinger equation. A fairly general result is obtained dividing particle velocities and corresponding wall motions into three regions: 1) low velocities, where motion is stochastic, ii) intermediate - islands of regular motion are imbedded in the stochastic sea, iii) high velocity - most trajectories are regular and bounded. In region ii) the KAM surfaces separate the regular islands from the stochastic background. The author claims that for certain ranges of momenta this model represents the Fermi accelerator quite well.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 35K10
zbMATH DE Number: 4144590
zbMATH Keywords: chaotic motion
zbMATH Keywords: time-dependent Schrödinger equation
zbMATH Keywords: new time variable
zbMATH Keywords: conserved energy
zbMATH Keywords: Floquet-Hamiltonian
zbMATH Keywords: KAM surfaces
zbMATH Keywords: Fermi accelerator
reviewed by: Vadim Komkov
MaRDI profile type: Publication
cites work: The quantized Baker's transformation
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full work available at URL: https://doi.org/10.1007/bf00399758
OpenAlex ID: W2045187139
description: scientific article; zbMATH DE number 4144590