Quantization of the Kramers and Pauli model

Abstract

The years ago Kramers and pauli quantized the diatomic molecular model which bears their names, obtaining the formula which has placed so large a part in hand spectroscopy. It is sometimes pointed out in modern text-books that these results, founded on the old mechanics, do not offer a very far-reaching analogy to those which the wave mechanics give for the real diatomic molecule. I think the reason will be seen on closely examining the classical calculation, either in the original papers, or in Born's "Atom-mechanik," Without adequate cause, all types of motion save one were rejected in the quantization; all that remained was the simple type of motion which, apart from the gyroscope's axial spin consists merely of a steady precession of the body as a whole about the invariable axis. It will here be shown that in reality we can have Kramers and Pauli models performing a very different type of motion, while completely satisfying the old-quantum equations (which were determined in the original papers). In these motions the gyroscope presesses not only about the invariable axis but also about the nuclear axis. Kemble proposed that in a modified model, having an elastically mounted gyroscope, notions of this more complex type should be possible as quantized States, though his actual treatment is confined to motions of the simpler type. As stated above, we shall see that the original model can have the more complex type of motion, in strictly quantized states. Kemble also proposed an energy formula for the corresponding motions of the real molecule; it is with this and with the well-established molecular formulæ of the wave mechanics that results obtained from the model are to be compared. For specifying orders of magnitude it is convenient to think what happens when the mass of the nuclei is made very great, their angular momentum retaining the same order of magnitude as that of the gyroscope. Then the existence of quantized motions other than those prescribed by Kramers and Pauli is not surprising; for by analogy with the vectorial conception of the real molecular (described, notably, by Hund and Weizel), we should expext, in these circumstances, to find models in whose quantized states the nuclei, if of great mass, will rotate about the invariable axis very s;lowly in comparison with the gyroscope's precession about the nuclear axis. For the real molecule, wave mechanics equations such as Kronig's (22) will hold with an accuracy increasing with the nuclear mass, in the sense that though the main rotational terms will became smaller, the neglected perturbations (if so they may be called) with diminish far more rapidly.