Establishing Quantum Physics in Munich by Michael Eckert

Author:Michael Eckert
Language: eng
Format: epub
ISBN: 9783030620349
Publisher: Springer International Publishing

Although Epstein’s recollection seems to dramatize his rivalry with Schwarzschild, it is by and large confirmed by the archival record. Both used methods from celestial mechanics. “I am very impressed that you cavort in Belgium and in quantum heaven at the same time,” Sommerfeld wrote in a response to a letter in which Schwarzschild, who was drafted for war service in Belgium, had reported how he approached the problem by using action-angle-variables. “Although I am not familiar with your notions from general celestial mechanics [...] I think that our views are not far apart from another.”25 Schwarzschild reformulated Sommerfeld’s quantization scheme by adapting the Hamilton-Jacobi formalism to quantum problems with the more general action integrals instead of Sommerfeld’s original phase integrals as the appropriate quantities for quantization. In a four-page letter dated March 21, 1916, he reported how he had solved the Stark effect problem.26 Sommerfeld responded three days later that Epstein had, at the same time, arrived at a “more general formula” that also contained a line missing in Schwarzschild’s result.27 Epstein’s and Schwarzschild’s preliminary communications were submitted for publication a week later (with one day difference). Schwarzschild died a few weeks later from a skin disease. Sommerfeld commemorated him in a paper titled “The Quantum Theory of Spectral Lines and the Last Work of Karl Schwarzschild” (Sommerfeld 1916b).

With Epstein’s and Schwarzschild’s work, the somewhat haphazard schemes of quantizing used by Bohr, Sommerfeld, and more recently, Planck (Eckert 2008) assumed a more coherent form that could be used to solve quantum problems. First, the equations of motion of the mechanical problem needed to be translated into the language of Hamilton-Jacobi formalism. The first (and most difficult) part of this procedure was to find out in which coordinates the classical problem was separable, specifically, to find the canonical form of the equations of motion. The “quantum” part of the problem was then fairly straightforward following Hamilton-Jacobi formalism. Sommerfeld first applied this procedure to the problem of the Zeeman effect. He had already indicated in his Annalen paper (Sommerfeld 1916a) that the plane orbit of an electron becomes spatially deformed by an applied magnetic field. The Hamilton-Jacobi approach to this problem led to phase integrals that were subject to complex integration—Sommerfeld’s specialty since his period in Göttingen as a mathematician. In September 1916, Sommerfeld submitted a paper to the Physikalische Zeitschrift (chosen for rapid publication to ensure high priority) where he presented this procedure for the Zeeman and the Stark effects, comparing both non-relativistic and relativistic motion. He also used the opportunity to simplify Epstein’s and Schwarzschild’s treatments of the Stark effect. With its focus on Hamilton-Jacobi formalism and complex integration, the paper presented concrete examples so that it could be taken as a role model for further applications (Sommerfeld 1916c).

Debye, who had in the meantime moved to Göttingen as director of the Physical Institute, published practically the same results on the Zeeman effect in the same issue of the Physikalische Zeitschrift. They were obtained by basically the same procedure and further emphasized this method of quantization (Debye 1916).

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