The trouble with orbits: The Stark effect in the old and the new quantum theory

Anthony Duncan, Michel Janssen

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14 Scopus citations


The old quantum theory and Schrödinger's wave mechanics (and other forms of quantum mechanics) give the same results for the line splittings in the first-order Stark effect in hydrogen, the leading terms in the splitting of the spectral lines emitted by a hydrogen atom in an external electric field. We examine the account of the effect in the old quantum theory, which was hailed as a major success of that theory, from the point of view of wave mechanics. First, we show how the new quantum mechanics solves a fundamental problem that one runs into in the old quantum theory with the Stark effect. It turns out that, even without an external field, it depends on the coordinates in which the quantum conditions are imposed which electron orbits are allowed in a hydrogen atom. The allowed energy levels and hence the line splittings are independent of the coordinates used but the size and eccentricity of the orbits are not. In the new quantum theory, this worrisome non-uniqueness of orbits turns into the perfectly innocuous non-uniqueness of bases in Hilbert space. Second, we review how the so-called WKB (Wentzel-Kramers-Brillouin) approximation method for solving the Schrödinger equation reproduces the quantum conditions of the old quantum theory amended by some additional half-integer terms. These extra terms remove the need for some arbitrary extra restrictions on the allowed orbits that the old quantum theory required over and above the basic quantum conditions.

Original languageEnglish (US)
Pages (from-to)68-83
Number of pages16
JournalStudies in History and Philosophy of Science Part B - Studies in History and Philosophy of Modern Physics
Issue numberPA
StatePublished - Nov 1 2014

Bibliographical note

Funding Information:
This paper was written as part of a joint project in the history of quantum physics of the Max-Planck-Institut für Wissenschaftsgeschichte and the Fritz-Haber-Institut in Berlin. The research of Anthony Duncan is supported in part by the National Science Foundation under Grant PHY-1212635 . The research of Michel Janssen is supported in part by the National Science Foundation under grant STS-1027018 .

Publisher Copyright:
© 2014 Elsevier Ltd.


  • Bohr-Sommerfeld theory
  • Epstein
  • Hamilton-Jacobi theory
  • Stark effect
  • WKB approximation
  • Wave mechanics


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