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)

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

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

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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

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