dorsal/arxiv
View SchemaExact solution of the Zeeman effect in single-electron systems
| Authors | A. Blom |
|---|---|
| Categories | |
| ArXiv ID | physics/0406141 |
| URL | https://arxiv.org/abs/physics/0406141 |
| DOI | 10.1088/0031-8949/2005/T120/014 |
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 field. 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.
{
"annotation_id": "d0d0db95-9f87-4273-a9af-61963178bcc1",
"date_created": "2026-03-02T18:00:50.374000Z",
"date_modified": "2026-03-02T18:00:50.374000Z",
"file_hash": "3c4ae0c0555d00371251f4e78523a4bfe41f4b95a0592f9e201ce3edb7af8108",
"private": false,
"record": {
"abstract": "Contrary to popular belief, the Zeeman effect can be treated exactly in\nsingle-electron systems, for arbitrary magnetic field strengths, as long as the\nterm quadratic in the magnetic field can be ignored. These formulas were\nactually derived already around 1927 by Darwin, using the classical picture of\nangular momentum, and presented in their proper quantum-mechanical form in 1933\nby Bethe, although without any proof. The expressions have since been more or\nless lost from the literature; instead, the conventional treatment nowadays is\nto present only the approximations for weak and strong fields, respectively.\nHowever, in fusion research and other plasma physics applications, the magnetic\nfields applied to control the shape and position of the plasma span the entire\nregion from weak to strong fields, and there is a need for a unified treatment.\nIn this paper we present the detailed quantum-mechanical derivation of the\nexact eigenenergies and eigenstates of hydrogen-like atoms and ions in a static\nmagnetic field. Notably, these formulas are not much more complicated than the\nbetter-known approximations. Moreover, the derivation allows the value of the\nelectron spin gyromagnetic ratio $g_s$ to be different from 2. For\ncompleteness, we then review the details of dipole transitions between two\nhydrogenic levels, and calculate the corresponding Zeeman spectrum. The various\napproximations made in the derivation are also discussed in details.",
"arxiv_id": "physics/0406141",
"authors": [
"A. Blom"
],
"categories": [
"physics.atom-ph",
"physics.plasm-ph"
],
"doi": "10.1088/0031-8949/2005/T120/014",
"title": "Exact solution of the Zeeman effect in single-electron systems",
"url": "https://arxiv.org/abs/physics/0406141"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "2dccd133-7a1d-4ccd-84af-e0a8e758c57d",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}