# From where did Heisenberg (1925) obtain the classical frequency equation?

In Quantum-Theoretical Re-Interpretation (1925), Heisenberg gives the following:

In order to characterize this radiation [of an electron] we first need the frequencies which appear as functions of two variables. In quantum theory these functions are of the form $$\nu(n,\alpha) = \frac{1}{h}\left[W(n)-W(n-\alpha)\right],$$ and in classical theory of the form $$\nu(n,\alpha) = \alpha\nu(n) = \alpha \frac{1}{h} \frac{dW}{dn}.$$

Here, $$\nu$$ is frequency, $$n$$ in the first equation is the quantum number of the first state, $$\alpha$$ is an integer such that the second state's quantum number is $$n-\alpha$$, and $$W$$ is the energy. Where does the classical equation come from, and how is $$n$$ to be interpreted? (It might be more specific to ask, how do we derive the classical equation from the correspondence principle?)

There, Bohr treats the problem of emission spectra by assuming only certain stationary states are allowed, each state being characterized by the quantum number $$n$$. The formula Heisenberg gives is not actually the classical formula, but the correspondence limit which (by the principle) should agree with classical physics. In particular, from Bohr in the reference cited:
On the present theory the frequency of the radiation emitted by the transition from the $$(n+1)$$th to the $$n$$th stationary state is equal to $$\frac{1}{h}(A_{n+1}-A_n)$$. When $$n$$ is large, this approaches to $$\frac{1}{h} \frac{dA_n}{dn}$$. On the ordinary electrodynamics we should expect the frequency of the radiation to be equal to the frequency of revolution, and consequently it is to be anticipated that for large values of $$n$$ $$\frac{dA_n}{dn} = h\omega_n.\quad .\ .\ .\ .\ .\ (8)$$
Now, by taking $$\alpha$$ to be the classical harmonic in the Fourier series representation of the (periodic) motion of the electron, we get $$\alpha \omega_n = \alpha \frac{1}{h}\frac{dA_n}{dn}$$ where $$A_n$$ denotes the energy in the $$n$$th state. .