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Michel Janssen

The Stark Effect in the Bohr-Sommerfeld Theory and in Schrödinger’s Wave Mechanics
Anthony Duncan and Michel Janssen, Pp. 217–271 in Finn Aaserud and Helge Kragh (Eds.), One Hundred Years of the Bohr Atom. Copenhagen: Det Kongelige Danske Videnskabernes Selskab, 2015.

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The explanation of the Stark effect in hydrogen, the splitting of the Balmer lines in an external electric field, was a major success of the old quantum theory of Bohr and Sommerfeld. Borrowing techniques from celestial mechanics, Epstein and Schwarzschild found frequencies for the Stark effect components of these lines that were in excellent agreement with Stark’s experimental data. Using Bohr’s correspondence principle, Kramers found the correct polarizations for these components and intensities that agreed, at least qualitatively, with the data. Shortly after the arrival of wave mechanics, Schrödinger and Epstein treated the Stark effect on the basis of the new theory. The two theories agree on the polarizations and, at least to first order in the strength of the external field, on the frequencies, but not on the intensities, where the new theory was soon found to be in reasonable quantitative agreement with new and better data. More importantly, the new theory eliminated the need for some additional assumptions that had to be made in the old theory. Furthermore, although this was not explicitly noted at the time, the new theory solved a fundamental problem in the old quantum theory that manifested itself glaringly in the Stark effect: it depends on the coordinates in which the quantum conditions are imposed which orbits are allowed. In the new theory, this worrisome non-uniqueness of orbits turns into the completely innocuous nonuniqueness of bases of eigenfunctions.