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A note on a remark by Landau regarding a charged particle in a magnetic field


Nagasawa, M (2002). A note on a remark by Landau regarding a charged particle in a magnetic field. Chaos, Solitons & Fractals, 14(7):1065-1070.

Abstract

In their famous book [Quantum mechanics: non-relativistic theory. Course of Theoretical Physics, Vol. 3, Translated from the Russian by J. B. Sykes and J. S. Bell, Pergamon Press Ltd., London, 1958; MR0093319 (19,1230k)], L. D. Landau and E. M. Lifshits explained a result of Landau on the motion of a charged quantum particle (an electron) in a homogeneous magnetic field (the z-axis being taken in the direction of the field), and claimed that the charged quantum particle (electron) makes clockwise rotation on the xy-plane. They however applied a gauge transformation to the vector potential to perform explicit computation, and as a result their arguments in justifying their claim became not transparent. Adopting the vector potential without the gauge transformation and applying my theory, I give a transparent proof for the claim of Landau-Lifshits, namely, the quantum particle moves on the xy-plane as a two-dimensional quantum harmonic oscillator with an additional drift of clockwise rotation. I remark moreover another important point that this clockwise rotation will continue after the magnetic field is switched off.

In their famous book [Quantum mechanics: non-relativistic theory. Course of Theoretical Physics, Vol. 3, Translated from the Russian by J. B. Sykes and J. S. Bell, Pergamon Press Ltd., London, 1958; MR0093319 (19,1230k)], L. D. Landau and E. M. Lifshits explained a result of Landau on the motion of a charged quantum particle (an electron) in a homogeneous magnetic field (the z-axis being taken in the direction of the field), and claimed that the charged quantum particle (electron) makes clockwise rotation on the xy-plane. They however applied a gauge transformation to the vector potential to perform explicit computation, and as a result their arguments in justifying their claim became not transparent. Adopting the vector potential without the gauge transformation and applying my theory, I give a transparent proof for the claim of Landau-Lifshits, namely, the quantum particle moves on the xy-plane as a two-dimensional quantum harmonic oscillator with an additional drift of clockwise rotation. I remark moreover another important point that this clockwise rotation will continue after the magnetic field is switched off.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2002
Deposited On:29 Nov 2010 16:27
Last Modified:05 Apr 2016 13:25
Publisher:Elsevier
ISSN:0960-0779
Publisher DOI:https://doi.org/10.1016/S0960-0779(02)00044-9
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1921777
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1041.81049

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