One hundred years ago, in July 1913, when the author was 28 years old, Philosophical Magazine received Niels Bohr's manuscript on his model of the atom. Happy birthday. Both 25-page papers are available in English here:
Bohr's model was essentially classical physics with the ad hoc assumption that the periodic orbits were restricted: \(\oint p\,dq\) had to be in \(2\pi\hbar\ZZ\). Quite accidentally, this assumption implied that the allowed total energy of the electron in an otherwise Keplerian orbit around the nucleus agrees with the energy extracted from the hydrogen emission and absorption spectrum (or from quantum mechanics that wasn't born yet).
Today, we know it was a coincidence. The correct spectrum must be extracted from quantum mechanics that was developed more than 10 years later (perhaps, they should have seen it faster?). The hydrogen atom happens to be sufficiently simple and solvable and its energy levels \(E_0/n^2\) just happen to agree with the levels that you obtain by the classical model restricted with a simple additional constraint.
The second Bohr's paper is dedicated to more complicated applications of Bohr's theory to other atoms and molecules. He should have seen that those things couldn't really work. Already the helium atom is a messy object and the classical three-body problem – with a nucleus and two electrons – is in some sense even messier than the quantum mechanical problem because the trajectories really don't want to be periodic.
We should also notice that Bohr had to overcome some additional steps towards quantum mechanics that was born in the mid 1920s. These steps went beyond the quantization of the orbital parameters. In particular, he had to realize that the transitions during which photons are emitted are governed by a nearly quantum logic. The electron that is changing its state of motion must be guaranteed that the new orbit will be allowed – it will obey the quantization rule for the orbital parameters. This requirement is almost directly extracted from the experimental data on spectroscopy but because it contradicts the classical logic so much (in classical physics, you would need an electron that plans the future, a mechanism that is sort of acausal, while the transition is perfectly causal in quantum mechanics), Bohr had to be courageous to propose it.
After some time, the correct theory was found and the reason why Bohr's theory looked good was understood. Some patience was needed. We should learn a lesson or two from these historical episodes.
Of course, many questions remained open even after 1925. For example, what is the glue that holds the nucleus together?
No, it has nothing to do with QCD or Gross or Wilczek or Politzer. Check the video above about the newest research which finally told us what is the answer. ;-) Via Honza U.