Discovery of the relativistic Schrödinger equation

K. Barley^a, J. Vega-Guzmán^b, A. Ruffing^c, S.K. Suslov^d
^aDepartment of Applied Mathematics and Statistics, Stony Brook University, New York, New York, USA
^bDepartment of Mathematics, Lamar University, 200 Lucas Building, P.O. Box 10047, Beaumont, TX, 77706, USA
^cLandeshauptstadt München, Referat für Bildung und Sport, Seminar TMG Universität, Drygalski Allee 2, 81477, München, 11794-3600, Germany
^dSchool of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, AZ, 85287-1804, USA

We discuss the discovery of the relativistic wave equation for a spin-zero charged particle in the Coulomb field by Erwin Schrödinger (presumably during the Christmas holidays of 1925—1926). However, in this new approach, an essential discrepancy was found with the fine structure formula for energy levels already obtained by Sommerfeld in the framework of the 'old' quantum mechanics. As a result, Schrödinger had to withdraw the original 'relativistically framed' article, a draft of which has never been found, from a journal and start all over with his centennial article on the nonrelativistic stationary Schrödinger equation. Our goal here is to follow the original 'relativistic idea' from a modern mathematical viewpoint and elaborate on why Schrödinger did not publish it. We hope that this consideration will encourage readers to study quantum physics at one of the crucial moments of its creation.

Keywords: Schrödinger equation, Klein—Fock—Gordon equation, Dirac equation, hydrogen atom, Sommerfeld fine structure formula, Wentzel—Kramers—Brillouin (WKB) approximation, Laplace transform, Nikiforov—Uvarov method
PACS: 01.65.+g, 03.65.−w (all)
DOI: 10.3367/UFNe.2021.06.039000
URL: https://ufn.ru/en/articles/2022/1/e/
Web of Science

000788597000005
Scopus

2-s2.0-85128558753
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2022PhyU...65...90B
Citation: Barley K, Vega-Guzmán J, Ruffing A, Suslov S K "Discovery of the relativistic Schrödinger equation" Phys. Usp. 65 90–103 (2022)

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PT Journal Article
TI Discovery of the relativistic Schrödinger equation
AU Barley K
FAU Barley K
AU Vega-Guzmán J
FAU Vega-Guzmán J
AU Ruffing A
FAU Ruffing A
AU Suslov S K
FAU Suslov SK
DP 10 Jan, 2022
TA Phys. Usp.
VI 65
IP 1
PG 90-103
RX 10.3367/UFNe.2021.06.039000
URL https://ufn.ru/en/articles/2022/1/e/
SO Phys. Usp. 2022 Jan 10;65(1):90-103

Received: 23rd, January 2021, revised: 21st, June 2021, accepted: 30th, June 2021

Оригинал: Барли К, Вега-Гузман Х, Руффин А, Суслов С К «Открытие релятивистского уравнения Шрёдингера» УФН 192 100–114 (2022); DOI: 10.3367/UFNr.2021.06.039000

E. Shrödinger is author of Physics-Uspekhi