Does the correspondence principle already cover this connection between classical orbits and quantum transitions? [duplicate]

I was contrasting the classical orbit of a planet with hydrogen energy levels. In the former case, change of energy leads to a continuous change of orbit (for example $a=-\frac{GMm}{2E}$), whereas in the latter case, there is a discrete change in levels ($h\nu=E_f-E_i$). It appears as if both these could be treated as a system changing states $S=(E,L)$, continuously in one case and discretely in another, based on whether action $S$ is much larger or of order of $\hbar$ – the first case being classical while the second case being quantum. Is this just the correspondence principle/Bohr-Sommerfeld quantization reworded? If so, could you provide me with a textbook/paper that derives this relationship using action-angle formalism or WKB approximation explicitly?