If we had not an atom, but an orbiting charged macroscopic system, we know from classical electromagnetism that the system would radiate electromagnetic energy at a frequency given by one divided by the period of the motion
where we have used (4) to find the period in terms of r.
On the other hand, in our quantum picture, the system emits photons of energy when making transitions between neighboring energy levels at spacing .
In the classical limit , we expect these two pictures to correspond so that the frequencies expected in both the classical and equation pictures agree. This gives us a very general result, relating the spacing in the quantum energy spectrum , to the classical period of the motion ,
This result is so general because we can imagine placing a small test charge on any system. As the energy increases the classical period gets larger and the quantum spacing gets less. More precisely,
So as and thus
And, in accordance with (11) and using (10) we find this is the same as