On the verge of Umdeutung 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 paper by John H. Van Vleck 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 shall derive is the Kramers dispersion formula, which played a key role in the thinking that led to Heisenberg's "Umdeutung" paper. Closely following Van Vleck's pre-Umdeutung approach, we derive classical and construct quantum formulae for the dispersion, emission, and absorption of radiation both for the special case of a charged harmonic oscillator (sec. 5) and for arbitrary non-degenerate multiply-periodic systems (sec. 6). In sec. 7, we rederive the same results using modern quantum mechanics. In sec. 8 we bring together the main conclusions of our study.