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arXiv:physics/0008050

Quasiclassical Calculations in Beam Dynamics

Antonina N. Fedorova, Michael G. Zeitlin

Published Aug 13, 2000• 0 citations
arXivHTMLPDF

Abstract

We present some applications of general harmonic/wavelet analysis approach (generalized coherent states, wavelet packets) to numerical/analytical calculations in (nonlinear) quasiclassical/quantum beam dynamics problems. (Naive) deformation quantization, multiresolution representations and Wigner transform are the key points.
DOI
10.48550/arxiv.physics/0008050

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We present the application of the variational-wavelet analysis to the quasiclassical calculations of the solutions of Wigner/von Neumann/Moyal and related equations corresponding to the nonlinear (polynomial) dynamical problems. (Naive) deformation quantization, the multiresolution representations and the variational approach are the key points. We construct the solutions via the multiscale expansions in the generalized coherent states or high-localized nonlinear eigenmodes in the base of the compactly supported wavelets and the wavelet packets. We demonstrate the appearance of (stable) localized patterns (waveletons) and consider entanglement and decoherence as possible applications.
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In the field of atomic and solid state physics, wavelet analysis has been applied so far in three different directions: (i) time-frequency analysis of harmonic generation in laser-atom interactions; (ii) ab initio electronic structure calculations in atoms and molecules; and (iii) construction of localized bases for the lowest Landau level of a 2-D electron gas submitted to a strong magnetic field. We survey these three types of applications, with more emphasis on methods than on precise results. Introduction There are two ways in which wavelets could play a role in atomic physics and possibly in solid state physics. First one may envisage them as physical objects , namely quantum states or wave functions. It is commonplace to remark that coherent states (CS) have a privileged role in atomic physics. Laser-atom interactions, revival phenomena, Rydberg wave packets and various semi-classical situations are all instances in which a coherent state description is clearly well-adapted. Of course, what is implied here are canonical CS, associated to the harmonic oscillator or the electromagnetic field [36]. But wavelets are also coherent states, namely those associated to the affine groups in various space dimensions, as we have seen in Chapter 2 (see [1] for a review on coherent states). Thus wavelets could well be thought of as convenient substitutes for canonical CS. However, this suggestion is still speculative at the present moment, very little has been achieved in this direction.
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