# Electron model under Maxwell's theory

I was not able to recall my memories, so:

What is the formula that states the frequency of electrons revolving around nucleus is equal to the frequency of light (or photon) emitted (or radiated)?

(I am of course talking of Maxwell's theory; in reality, we know that this is not true.)

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It is true for Rydberg atoms, and close enough to true in quantum systems to allow Bohr to find the spectrum. There is no formula for this--- it is just obvious that a time varying source with a given frequency produces light with the same period. –  Ron Maimon Jun 26 '12 at 3:34

The rule is that if you have a classical source with frequency f, the outgoing radiation is a superposition of frequencies f,2f,3f, etc, according to the Fourier decomposition of the source frequency.

If you write Maxwell's equation in Lorentz gauge:

$$\partial_\mu \partial_\mu A = J$$

And assume that J is periodic, by Fourier tranform:

$$k^2 A(k,\omega) = J(k,\omega)$$

So that the fourier transform of A is supported on the same frequencies as J. This is obvious--- a periodic source gives rise to a periodic wave with the same period.

The rule is correct quantum mechanically too in the correspondence limit: the emission between level n and n-k is at a frequency which is k times the inverse classical orbital frequency at level n. You can see how Borh used this to derive the quantization rule here: Bohr Model of the Hydrogen Atom - Energy Levels of the Hydrogen Atom .

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I am telling you the part that everyone leaves out of books. How did Bohr know how you are supposed to quantize? Did he make up the condition $L=n\hbar$ out of thin air? He assumed that the frequency of radiation would be at the classical frequency for a large circular orbit, then this tells you what the distance in energy between neighboring orbits is: it's one quantum at the classical orbital frequency. This argument is correct to leading order, I review it here: physics.stackexchange.com/questions/28520/… . –  Ron Maimon Jun 26 '12 at 7:39