About electron radiation frequency in Heisenberg's 1925 paper

Question

In Heisenberg's 1925 article Quantum Theoretical Interpretation of Kinematic and Mechanical Relations, one of the first things he establishes are the form of the frequency functions in (what I assume are) energy transitions: \begin{align*}& \nu(n,n-\alpha)=\frac{1}{h}[W(n)-W(n-\alpha)] \\ &\nu(n,\alpha)=\alpha\nu(n)=\frac{\alpha}{h} \frac{dW}{dn}\end{align*}

where the first relation corresponds to the quantum case and the second to the classical.

I think I recognize the first relation if $W$ is the energy of the energy levels (probably still reminiscent of Einsteins work function?). But, in that case, where does the classical expression come from? My knowledge on radiation theory is very limited, very basic stuff from an electrodynamics undergraduate course. If this is out of the scope of an answer I would be grateful if anyone could give me a reference instead.

Answer 1

It's become obscure over time, but what he's specifically referencing is Bohr's correspondence principle. The word 'classical' is a bit confusing here. The frequency function comes from the fact that in the pre-Bohr theory, the energy levels aren't quantized, so $W$ can take on a range of continuous values.

It doesn't come from classical electromagnetics, because in classical electromagnetics there's no relation between frequency and energy. The Planck-Einstein equation $E = hv$ comes from quantum theory.

Bohr wrote about this correspondence principle in fragments over several papers, for example in On the Quantum Theory of Line Spectra (part I, page 15). More info given here.