Bohr-Sommerfeld quantum condition

Bonino brought forward a further contribution to the theory of infrared spectra of organic liquids by incorporating the Bohr-Sommerfeld quantum conditions, including the correspondence principle of Bohr as well. This paved the way toward establishing a correlation between the physical and chemical image of molecules in the study of infrared spectra. From this series of papers on infrared spectroscopy, one can already observe the interdisciplinary character of Bonino s thought. In a lecture delivered some years later, Bonino offered these reflections on his chosen field of research ... 

Keller, J.B. (1958). Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Ann. Phys. 4, 180-188. 

In Bohr s model of the hydrogen atom, the circular orbits were determined by the quantum number more accurately, by the square of the quantum number n. No other orbits were allowed. By changing the orbits from circles to ellipses, Sommerfeld introduced a second radius, which gave him another variable to play with. So it was that Sommerfeld generalized Bohr s quantum condition for electron orbits in terms of the two quantum numbers n and k. His analysis led him to establish a relationship between the two quantum numbers namely, the quantum number n set the upper limit on the quantum number k, but k could have smaller values as follows ... 

For a diatom (as for a separable vibrational mode in a polyatomic) the product vibrational quantum number is found from the Bohr-Sommerfeld quantization conditions namely that pr dr = (v 4- 1/2)h for bound motions (27). That is, if the momentum is followed over one half-period the product vibrational action can be calculated ... 

This tremendous accomplishment of Bohr stimulated other theoreticians, and soon Sommerfeld (1916) had generalized Bohr s quantum conditions and applied the results to elliptical orbits in the hydrogen a tom. By including relativistic effects he was able to interpret many of the details of the hydrogen fine structure. The introduction of an additional quantum number describing the allowed orientations... 

This equation resembles the Bohr-Sommerfeld condition pdx = nh, but differs from it in that S now is the solution of the quantum HJ equation and not of the classical one as before. 

Bohr applied a quantum condition to the energy states of the hydrogen atom only certain energy states were allowed. Sommerfeld applied a quantum condition to the orientation of electron orbits only certain spatial orientations relative to an applied magnetic field were allowed. The experiment of Stem and Ger-lach was designed to test Sommerfeld s explanation of the Zeeman effect, namely, the idea of space quantization. 

The strongest pieces of evidence complex atoms provide in favour of independent electron modes and simple Bohr-Sommerfeld quantisation are (i) the existence of Rydberg series and (ii) the regularity of the periodic table of the elements. As a corollary, we should look for quantum chaos (if it occurs) in atoms for which there is some breakdown in the quality of the shell structure, combined with prolific and heavily perturbed overlapping series of interacting levels. These conditions are most readily met, as will be shown below, in the spectra of the alkaline-earth elements, as a result of d-orbital collapse. 

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