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A note on a remark by Landau regarding a charged particle in a magnetic field. (English) Zbl 1041.81049

Summary: In the famous book by Landau-Lifshitz Quantum Mechanics (1965; Zbl 0178.57901), they 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 clock-wise 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 his theory, the author will give a transparent proof for the claim of Landau-Lifshitz, namely, the quantum particle moves, on the \(xy\)-plane, as a two-dimensional quantum harmonic oscillator with additional drift of clock-wise rotation. The author moreover remarks another important point that this clock-wise rotation will continue after the magnetic field will be switched off.

MSC:

81Q50 Quantum chaos
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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References:

[1] Landau, L. D.; Lifshitz, E. M., Quantum mechanics, non-relativistic theory (1958/1965), Pergamon Press: Pergamon Press Oxford London · Zbl 0178.57901
[2] Nagasawa, M., Schrödinger equations and diffusion theory (1993), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0780.60003
[3] Nagasawa, M., Stochastic processes in quantum physics (2000), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0998.81519
[4] Nagasawa M. On quantum particles. Chaos, Solitons & Fractals 13 (2002) 1393-1405; Nagasawa M. On quantum particles. Chaos, Solitons & Fractals 13 (2002) 1393-1405 · Zbl 0994.81021
[5] Schrödinger E. Über die Umkehrung der Naturgesetze. Sitzungsberichte der preussischen Akad. der Wissenschaften Physikalisch Mathematische Klasse, 1931. p. 144-53; Schrödinger E. Über die Umkehrung der Naturgesetze. Sitzungsberichte der preussischen Akad. der Wissenschaften Physikalisch Mathematische Klasse, 1931. p. 144-53 · Zbl 0001.37503
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