Quantum Trajectory Description of the Time-Independent (Inverse) Fermi Accelerator
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ATOMIC PHYSICS
Quantum Trajectory Description of the Time-Independent (Inverse) Fermi Accelerator M. S. Hussein1,2,3 · B. Poirier4,5 Received: 15 October 2020 / Accepted: 6 November 2020 © Sociedade Brasileira de F´ısica 2020
Abstract We re-examine the (inverse) Fermi accelerator problem by resorting to a quantum trajectory description of the dynamics. Quantum trajectories are generated from the time-independent Schr¨odinger equation solutions, using a unipolar treatment for the (light) confined particle and a bipolar treatment for the (heavy) movable wall. Analytic results are presented for the exact coupled two-dimensional problem, as well as for the adiabatic and mixed quantum-classical approximations. Keywords Fermi accelerator · Quantum trajectories · Born-Oppenheimer approximation · Mixed quantum-classical · Chemical physics · Nuclear physics · Quantum dynamics · Quantum friction · Energy dissipation
1 Introduction During the 1990s, one of the authors (Hussein), working on problems connected with the idea of friction in quantum mechanics, came to realize that an old model devised by Enrico Fermi could have benefits in an entirely different context. Fermi originally developed his accelerator model to explain the energy gain by particles traversing plasmas (clouds of charged particles), of interest in cosmic ray physics [1–3]. Hussein, a nuclear physicist, realized that a very similar model could be used to exploit energy loss in quantum systems. This would then lead to a microscopic, albeit simplified, derivation of what may be called the “quantum friction force” [4–6]. B. Poirier
[email protected] 1
DCTA, Instituto Tecnol´ogico de Aeron´autica, S˜ao Jos´e dos Campos, SP, 12.228-900 Brazil
2
Instituto de Estudos Avanc¸ados, Universidade de S˜ao Paulo, C. P. 72012, S˜ao Paulo, SP, 05508-970, Brazil
3
Instituto de F´ısica, Universidade de S˜ao Paulo, C. P. 66318 S˜ao Paulo, SP, 05314-970, Brazil
4
Max-Planck-Institut f¨ur Physik Komplexer Systeme, N¨othnitzerstraße. 38, D-01187 Dresden, Germany
5
Department of Chemistry and Biochemistry, Texas Tech University, 2500 Broadway, Box 41075 Lubbock, TX 79409-1061, USA
In fact, friction forces were in use in nuclear physics research during the 1970s and 1980s, in connection with what came to be known as “deep inelastic collisions” (DIC), where two colliding nuclei experience a great amount of kinetic energy loss [7–11]. At that time, nuclear theorists devised statistical models to deal with reactions of this type that resulted in Fokker-Planck equations and friction forces. Hussein’s contribution was to analyze the much simpler problem of the inverse Fermi accelerator, involving a particle colliding between two walls, one of which is movable (see Fig. 1). Over time, an exchange of energy ensues; the particle loses energy as the wall gains it. This amounts to an effective friction force that acts on the particle and depends on the ratio of the small particle mass m to that of the greater wall mass M. In an Annals of Physics article [4], M. Hussein and
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