On the verge of <Emphasis Type="Italic">Umdeutung</Emphasis> in Minnesota: Van Vleck and the correspondence principle. Part two

This is the second installment of a two-part paper on developments in quantum dispersion theory leading up to Heisenberg’s Umdeutung paper. In telling this story, we have taken a 1924 paper by John H. Van Vleck in The Physical Review as our main guide. In this second part we present the detailed derivations on which our narrative in the first part rests. The central result that we derive is the Kramers dispersion formula, which played a key role in the thinking that led to Heisenberg’s Umdeutung paper. We derive classical formulae for the dispersion, emission, and absorption of radiation and use Bohr’s correspondence principle to construct their quantum counterparts both for the special case of a charged harmonic oscillator (Sect. 5) and for arbitrary non-degenerate multiply-periodic systems (Sect. 6). We then rederive these results in modern quantum mechanics (Sect. 7). 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 authors gratefully acknowledge support from the Max Planck Institute for History of Science. The research of Anthony Duncan is supported in part by the National Science Foundation under grant PHY-0554660.