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Electron spin and probability current density in quantum mechanics

Kerr, William C.

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abstract
This paper analyzes how the existence of electron spin changes the equation for the probability current density in the quantum-mechanical continuity equation. A spinful electron moving in a potential energy field experiences the spin-orbit interaction, and that additional term in the time-dependent Schrödinger equation places an additional spin-dependent term in the probability current density. Further, making an analogy with classical magnetostatics hints that there may be an additional magnetization current contribution. This contribution seems not to be derivable from a non-relativistic time-dependent Schrödinger equation, but there is a procedure described in the quantum mechanics textbook by Landau and Lifschitz to obtain it. We utilize and extend their procedure to obtain this magnetization term, which also gives a second derivation of the spin-orbit term. We conclude with an evaluation of these terms for the ground state of the hydrogen atom with spin-orbit interaction. The magnetization contribution is generally the larger one, except very near an atomic nucleus.
subject
current density
spin orbit interactions
wave functions
angular momentum
textbooks
citation
7 (issue)
82 (volume)
contributor
Hodge, W. B. (author)
Migirditch, S. V. (author)
Kerr, William C. (author)
date
2014-07-30T15:57:44Z (accessioned)
2014-07-30T15:57:44Z (available)
2014 (issued)
identifier
Hodge, W.B.; Migirditch, S.V.; & Kerr, W.C. (2014). "Electron spin and probability current density in quantum mechanics." American Journal of Physics 82(7):681-690. Available at http://dx.doi.org/10.1119/1.4868094. (citation)
http://dx.doi.org/10.1119/1.4868094 (doi)
http://hdl.handle.net/10339/39352 (uri)
publisher
American Association of Physics Teachers
rights
Copyright 2014 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Am. J. Phys. 82, 681 (2014) and may be found at http://dx.doi.org/10.1119/1.4868094. (holder)
source
American Journal of Physics
title
Electron spin and probability current density in quantum mechanics
type
Article

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