What lead N.Bohr to come up with his postulates?

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I can find information everywhere about his postulates but not how he came up with them either than the hydrogen spectrum.
 

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vanhees71
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The usual story is that Bohr met Rutherford when he made is famous gold-foil experiment which demonstrated that atoms are made of a tiny positively charged nucleus with the electrons somehow grouped around it. In between there's just a lot of empty space. That picture cannot be right within classical physics, because it implies that the electrons must run around the nucleus like the planets run around the sun. The difference between this analogous systems is that in the atom it's the electromagnetic interaction that keeps the system together, and this is by a factor of about ##10^{40}## larger than the gravity holding the solar system together. Now relativity implies that both systems must radiate off electric or gravitational waves, respectively. While the gravitational waves are very weak, and it takes ages to radiate off enough energy from the planets running around the sun that it is negligible over the time range we look at them, for the atom its a very short time, where the electron would radiate off all it's energy and consequently must crash into the nucleus. In short: Classical physics inevitably predicts the unstability of the hydrogen atom, which is of course completely against experience.

Now Bohr introduced ad hoc assumptions on the stability of certain orbits by making up his "quantum rule", according to which the phase-space area of any action-angle variable pair must be an integer multiple of ##\hbar/(2 \pi)## (Plancks quantum of action). This worked beautifully for the hydrogen atom in the sense that now you have these stable orbits and only when the electron jumps from a higher energetic orbit to a lower energetic one it radiates off a photon with the energy difference of the energy levels as its energy ##E_{\gamma}=\hbar \omega## with ##\omega## the frequency of the photon. This lead to the correct description of the hydrogen spectrum (neglecting the fine structure).

Then Sommerfeld refined the model and making it also relativistic. The funny thing is that he got even the correct fine structure!

The flaw, however is that all this is an accident, working well for the hydrogen atom only, and this must have to do with the large symmetry of the Kepler problem, which seems to make the theoretical treatment very robust, i.e., nearly any theory can predict the correct energy levels. However, when the physicists at the time applied the methods of this so discovered "old quantum theory" they got frustrated since although they got rough qualitative agreement also for multi-electron atoms (using Pauli's exclusion principle as one more assumption), but to get it quantitatively right, they had to introduce more and more ad hoc assumptions for any sort of atom. That's not very convincing, and indeed, with the advent of modern quantum theory the Bohr-Sommerfeld model was history (and should, in my opinion, remain there and not be taught in great detail anymore since it's more confusing and hindering the understanding of modern QT than it helps in any way).
 
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