Some recent work has been done by Back and Goudsmidt on the “hyperfine” structure of the optical spectrum of bismuth,* and more recently similar work has been carried out for cæsium by Jackson. In each of these investigations the line structure was examined closely with a view to revealing a still finer structure, and it was found in both cases that the lines attributed to electronic spin were themselves composed of several distinct lines. In fact, for cæsium, each of the fine (electron spin) lines of the principal series was found to split up into two ; for bismuth the hyperfine structure was more complicated. Back and Goudsmidt attributed the structure to a nuclear spin, and working out the consequences of this on the lines of the old quantum mechanics they found that a nuclear spin of 41/2 quanta is necessary to account for the facts ; a spin of a 1/2 quantum is similarly attributed by Jackson to the nucleus of cæsium. The hypothesis explains very satisfactorily in a qualitative way the results of observation. In the work described in the present paper the methods of the new quantum mechanics have been applied to the problem. More precisely, we consider the motion of a single electron in a Coulombian field due to a nucleus possessing a 1/2 quantum of spin. It will be seen that the results can easily be extended to the case of any central field, and the principle could also be extended to the case of an atom with a nuclear spin of 1/2(
nh
/2π), but the detailed working out would be very heavy for n > I (at any rate, using the methods explained in this paper), owing to the large number of wave functions which would be necessary to specify any state of the atom. It will be seen that the results we obtain are substantially the same as Jackson’s so far as the energy levels are concerned, but the calculated intensities are not consistent with the observed transitions, and we deduce a combination rule which is radically different from Jackson’s.
Quantum SU(2|1) supersymmetric ℂN Smorodinsky-Winternitz system
