It is possible and the general case was discussed by Bohr and Sommerfeld. There's no difference between the ``old quantum theory'' and ``quantum mechanics'' but history. and a great deal of simplification and clarification-quantum mechanics is much simpler than the ``old quantum theory''. In fact it's the other way around, as Fock and Pauli found in 1926: It's possible to describe the Bohr model in terms of the solution to the harmonic oscillator and that's the correct way to understand that the spectrum of a quantum particle in the Coulomb potential is bounded from below.

no, both quantum mechanics and old quantum theory introduce good discussion for Bohr atom using harmonic oscillator scenario

Bohr could not deal with the harmonic oscillator because he quantized angular momentum and there is no angular momentum in a linear oscillator.

You can find solutions to the harmonic oscillator using the Sommerfeld theory, only the trouble is you get the wrong answer, which is probably why they don't reproduce it in text books. The Sommerfeld approach quantized the action integral per period, i.e. the quantum condition was nh, but ran into the problem that the quantum number n = 0 was a legitimate solution because 0 is an integer, whereas for the Schrödinger equation the quantization condition is (n + 1/2). The old quantum theory permitted the ground state to be stationary whereas there has to be zero point energy.

You might notice that circular motion can be described as two harmonic oscillators because it is effectively two degrees of freedom, and for the Schrödinger equation we have (nr + 1/2) + (ℓ + 1/2), hence n cannot get below 1. For Bohr, n = 0 was arbitrarily stated as not possible, yet it should occur according to Maxwell with Bohr's quantum condition.

Yes, zero point energy of 1-d oscillator is an important omission of old quantum theory. Whereas, a 2-d oscillator can be dealt with correctly. Qualitatively, old theory did not have concept of tunnelling. Once you accept wave-like properties for particles, they can penetrate into the barrier. Further, there is another observation. For circular Bohr orbits, and for particle in a box model, you can fit integral number of wavelengths into the orbit. But, it is no longer possible with the elliptical orbits of Hydrogen, nor with 1-d oscillator.

This can be found exactly if you modify the Bohr-Sommerfeld quantization conditions slightly i.e. assume that the closed phase-space integral Closed_Integral p dx = 2 Pi hbar (n +1/2) but not 2 Pi hbar n i.e different from the original only by the electron spin or if there is some zero-n residue in it. For the harmonic oscillator the integral p dx over one harmonic oscillator period between the turning points V(x_i) = E (back and forth from the one turning point) can be readily calculated as simply the area integral under the semicircle of the radius proportional to the square root of the energy E (y(x) = (r^2 - x^2)^0.5 is the equation of the semicircle and so clearly Integ y(x)_x1^x2 = Pi r^2/2, y(xi) = 0) i.e.proportional to the area pi (E^0.5)^2/2 = Pi E /2 or 2 Integ_x1^x2 p dx = 2 Integ_x1^x2 Sqrt(2 m (E_n - V(x))) dx 2 Pi hbar (n+1/2) where V(x) = m omega^2/2 from what E_n = (n + 1/2) hbar omega can be obtained. It is actually quite funny that the correct BS quantizations are different for the hydrogen atom and for the harmonic oscillator to obtain the correct energy spectra but using the wrong condition to the Bohr atom gives wrong energy spectrum for low n but in some sense predicts the electron spin when it allows n=0 and the lowest angular momentum of the Hydrogen hbar/2 and it gives the correct energy spectrum of the Hydrogen atom in only 2-dimensions E_n=-1/(n+1/2)^2/2 Article The two-dimensional hydrogen atom revisited

Matt Kalinski

What you say will get you the right answer but I don't think there is any realistic way Sommerfeld could get to it because he had to assume the periodic time of an oscillation was twice that (to get the 1/2). We can do what you say now but we know the answer. Basically, you have to say it is impossible to stop the oscillation, i.e. a stationary object is impossible and in classical mechanics that is just not right, and in Sommerfeld's time there was no reason to think there was any reason not to permit the stationary object, unless they really asked why ground state hydrogen was stable. The circular orbit could be considered as two harmonic oscillations at right angles, in which case the stable ground state hydrogen would require a 1/2 for one such oscillator. Sommerfeld would have to have been fairly good to get that though because nobody else did.

Yet another example of the new being the well forgotten old!

The Bohr-Sommerfeld quantization condition of the "old quantum mechanics" follows from the Schroedinger equation within the WKB approximation. It is derived, for instance, by Landau and Lifshitz in Theoretical Physics, v. III, Quantum Mechanics, part VII. Therefore we can say that anything treatable by the Schroedinger equation is also doable by the old QM. Another matter that WKB normally screws up the low-lying levels. That it works so well for the Hydrogen atom starting from the ground state is rather an incident.

Caveat: Traditional WKB works only for 1D systems and systems with separable variables. Generalization of WKB beyond these simple cases was achieved relatively recently. Generalized Bohr-Sommerfeld conditions involve the so-called topological indices (don't ask me what they are). For a pedagogical approach, see part V in http://chaosbook.org/chapters/ChaosBook.pdf.