This article is cited in 1 paper

FROM THE HISTORY OF PHYSICS. THE 100th ANNIVERSARY OF QUANTUM MECHANICS

Old quantum mechanics by Bohr and Sommerfeld from modern perspective

K. Barley^a, A. Ruffing^b, S. K. Suslov<sup>c

a</sup> Department of Mathematics, College of Arts and Sciences, Howard University, Washington, USA
^b Landeshauptstadt München, Seminar TMG Universität, Referat für Bildung und Sport, München, Germany
^c School of Mathematical and Statistical Sciences, Arizona State University, Tempe, USA

Abstract: 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.

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

Received: July 3, 2025
Accepted: August 15, 2025

DOI: 10.3367/UFNr.2025.08.040014

 English version: DOI: 10.3367/UFNe.2025.08.040014