Old quantum mechanics by Bohr and Sommerfeld from modern perspective

K. Barley^†^a, A. Ruffing^‡^b, S.K. Suslov^§^c
^aDepartment of Mathematics, College of Arts and Sciences, Howard University, 2141 6th St NW, Washington, DC, 20059, USA
^bLandeshauptstadt München, Seminar TMG Universität, Referat für Bildung und Sport, Marienplatz 8, München, 80331, Germany
^cSchool of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, AZ, 85287-1804, USA

We review Bohr's atomic model and its extension by Sommerfeld from a mathematical perspective of wave mechanics. The derivation of quantization rules and energy levels is revisited using semiclassical methods. Sommerfeld-type integrals are evaluated by elementary techniques, and connections with the Schrödinger and Dirac equations are established. Historical developments and key transitions from classical to quantum theory are discussed to clarify the structure and significance of the old quantum mechanics.

Fulltext pdf (1.4 MB)
Appendix: AppendixEeng.pdf pdf (208 KB)
Appendix: EllipsesAnimateSofisticated.nb nb (202 KB)
Appendix: EllipsesAnimateAu.nb nb (120 KB)
Appendix: AppendixE.nb nb (76 KB)
Appendix: BohrAtomMathematica.nb nb (1.5 MB)
Appendix: S.K. Suslov. Remarcable Dates and Place: One Hundred Years Ago pdf (12 MB)

	By downloading this file I agree with the terms of use.
	Reproduction of material from the journal in any form requires written permission of the publisher. Teachers, students and researchers can individually use the materials on Physics-Uspekhi website for noncommercial use in a teaching or research activities.

Fulltext is also available at DOI: 10.3367/UFNe.2025.08.040014

Correspondence should be addressed to
Keywords: Bohr's atomic model, Bohr—Sommerfeld quantization rule, Schrödinger equation, Dirac equation, WKB method, fine structure formula, Sommerfeld puzzle, computer algebra system: Mathematica, transuranic elements
PACS: 01.65.+g, 02.70.−c, 03.65.−w (all)
DOI: 10.3367/UFNe.2025.08.040014
URL: https://ufn.ru/en/articles/2026/1/e/
Web of Science

001671153400004
Scopus

2-s2.0-105031182709
ADS NASA

2025arXiv250600408B
Citation: Barley K, Ruffing A, Suslov S K "Old quantum mechanics by Bohr and Sommerfeld from modern perspective" Phys. Usp. 69 74–93 (2026)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Received: 3rd, July 2025, accepted: 15th, August 2025

Оригинал: Барли К, Руффинг А, Суслов С К «Старая квантовая механика Бора и Зоммерфельда с современной точки зрения» УФН 196 83–103 (2026); DOI: 10.3367/UFNr.2025.08.040014

References (116) Cited by (1) Similar articles (20) ↓

K. Barley, Vega-Guzmán Jose et al “Discovery of the relativistic Schrödinger equation” Phys. Usp. 65 90–103 (2022)
V.P. Milant’ev “Creation and development of Bohr’s theory (on the 90th anniversary of the Bohr theory of the atom)” Phys. Usp. 47 197–203 (2004)
S.V. Petrov “Uniqueness of spin 1/2” Phys. Usp. 69 94–98 (2026)
V.S. Popov “"The host of heaven is far..." (my recollections of Ya B Zeldovich)” Phys. Usp. 57 413–418 (2014)
V.I. Sanyuk, A.D. Sukhanov “Dirac in 20th century physics: a centenary assessment” Phys. Usp. 46 937–956 (2003)
L.B. Okun “V A Fock and gauge symmetry” Phys. Usp. 53 835–837 (2010)
V.I. Kogan “The discovery of the Planck constant: ’roentgenoscopy’ of the scientific situation (1900). Missed opportunities in the choice of the Second Step (on the centenary of the First Step of quantum theory)” Phys. Usp. 43 1253–1259 (2000)
M.I. Monastyrskii “John von Neumann” Phys. Usp. 47 1271–1280 (2004)
V.Ya. Frenkel’ “YURII ALEKSANDROVICH KRUTKOV” Sov. Phys. Usp. 13 816–825 (1971)
V.L. Ginzburg “Tricentenary of Isaac Newton’s ’Mathematical Principles of Natural Philosophy’” Sov. Phys. Usp. 30 46–59 (1987)
G.B. Malykin, V.S. Savchuk, E.A. Romanets “Lev Yakovlevich Strum and the hypothesis of the existence of tachyons” Phys. Usp. 55 1134–1139 (2012)
V.A. Fock “On the invariant form of the wave and motion equations for a charged point mass” Phys. Usp. 53 839–841 (2010)
A. Sommerfeld “On the addition of velocities in the theory of relativity” Phys. Usp. 53 929–931 (2010)
G.I. Marchuk, V.S. Imshennik, M.M. Basko “The physics of a thermonuclear explosion of a normal-density liquefied deuterium sphere (On the impossibility of a spherically symmetric thermonuclear explosion in liquid deuterium at normal density)” Phys. Usp. 52 267–273 (2009)
Yu.Ya. Stavissky “Nuclear energy for space missions” Phys. Usp. 50 1179–1187 (2007)
M.V. Vol’kenshtein “Complementarity, physics, and biology” Sov. Phys. Usp. 31 140–150 (1988)
I.A. Shcherbakov “On the 60th anniversary of Nobel Prize for discovery of laser-maser principle” Phys. Usp. 67 845–848 (2024)
G.B. Malykin “Noncommutative nature of the addition of noncollinear velocities in special relativity and the geometric phase method (commemorating the publication centennial of A Sommerfeld’s work)” Phys. Usp. 53 923–927 (2010)
B. Pontecorvo “Pages in the development of neutrino physics” Sov. Phys. Usp. 26 1087–1108 (1983)
G.G. Denisov, A.G. Litvak, A.M. Sergeev “Andrey Viktorovich Gaponov-Grekhov: a brief essay on his life and scientific activity (to the 100th anniversary of his birth)” Phys. Usp. 69 (6) (2026)

The list is formed automatically.