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
