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Exact solution of the Zeeman effect in single-electron systems

Blom, Anders LU (2005) In Physica Scripta T120. p.90-98
Abstract
Contrary to popular belief, the Zeeman effect can be treated exactly in single-electron systems, for arbitrary magnetic field strengths, as long as the term quadratic in the magnetic field can be ignored. These formulas were actually derived already around 1927 by Darwin, using the classical picture of angular momentum, and presented in their proper quantum- mechanical form in 1933 by Bethe, although without any proof. The expressions have since been more or less lost from the literature; instead, the conventional treatment nowadays is to present only the approximations for weak and strong fields, respectively. However, in fusion research and other plasma physics applications, the magnetic fields applied to control the shape and position... (More)
Contrary to popular belief, the Zeeman effect can be treated exactly in single-electron systems, for arbitrary magnetic field strengths, as long as the term quadratic in the magnetic field can be ignored. These formulas were actually derived already around 1927 by Darwin, using the classical picture of angular momentum, and presented in their proper quantum- mechanical form in 1933 by Bethe, although without any proof. The expressions have since been more or less lost from the literature; instead, the conventional treatment nowadays is to present only the approximations for weak and strong fields, respectively. However, in fusion research and other plasma physics applications, the magnetic fields applied to control the shape and position of the plasma span the entire region from weak to strong fields, and there is a need for a unified treatment. In this paper we present the detailed quantum- mechanical derivation of the exact eigenenergies and eigenstates of hydrogen-like atoms and ions in a static magnetic. eld. Notably, these formulas are not much more complicated than the better-known approximations. Moreover, the derivation allows the value of the electron spin gyromagnetic ratio g(s) to be different from 2. For completeness, we then review the details of dipole transitions between two hydrogenic levels, and calculate the corresponding Zeeman spectrum. The various approximations made in the derivation are also discussed in details. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Physica Scripta
volume
T120
pages
90 - 98
publisher
Institute of Physics Publishing Ltd.
external identifiers
  • wos:000236907000016
  • scopus:42349083502
ISSN
0031-8949
DOI
10.1088/0031-8949/2005/T120/014
language
English
LU publication?
yes
id
ed237ed4-35d0-471d-8f24-74a5e801205c (old id 208459)
date added to LUP
2007-08-10 10:59:40
date last changed
2017-01-01 05:08:38
@article{ed237ed4-35d0-471d-8f24-74a5e801205c,
  abstract     = {Contrary to popular belief, the Zeeman effect can be treated exactly in single-electron systems, for arbitrary magnetic field strengths, as long as the term quadratic in the magnetic field can be ignored. These formulas were actually derived already around 1927 by Darwin, using the classical picture of angular momentum, and presented in their proper quantum- mechanical form in 1933 by Bethe, although without any proof. The expressions have since been more or less lost from the literature; instead, the conventional treatment nowadays is to present only the approximations for weak and strong fields, respectively. However, in fusion research and other plasma physics applications, the magnetic fields applied to control the shape and position of the plasma span the entire region from weak to strong fields, and there is a need for a unified treatment. In this paper we present the detailed quantum- mechanical derivation of the exact eigenenergies and eigenstates of hydrogen-like atoms and ions in a static magnetic. eld. Notably, these formulas are not much more complicated than the better-known approximations. Moreover, the derivation allows the value of the electron spin gyromagnetic ratio g(s) to be different from 2. For completeness, we then review the details of dipole transitions between two hydrogenic levels, and calculate the corresponding Zeeman spectrum. The various approximations made in the derivation are also discussed in details.},
  author       = {Blom, Anders},
  issn         = {0031-8949},
  language     = {eng},
  pages        = {90--98},
  publisher    = {Institute of Physics Publishing Ltd.},
  series       = {Physica Scripta},
  title        = {Exact solution of the Zeeman effect in single-electron systems},
  url          = {http://dx.doi.org/10.1088/0031-8949/2005/T120/014},
  volume       = {T120},
  year         = {2005},
}