Abstract 1. Indications of the occurrence of fractional quantum numbers have already been found by Kratzer from a study of certain band spectra, and by Curtis in his experimental investigation of helium bands. Heisenberg has also employed half-integral numbers in a quantum theory of the anomalous Zeeman effect with considerable success. The fourth of the slightly extended quantum conditions recently suggested by W. Wilson and applied by the present writer to the Zeeman effect involves a fractional quantum number, as O. W. Richardson has pointed out. K. F. Niessen, in his work on the positively ionised hydrogen molecule, observes certain discontinuities in the energy graphs between symmetrical and asymmetrical models which strongly the occurrence of half-integral quantum orbits for the latter. He refrains, however, from making this assumption, on the ground that it would necessitate the adoption of half numbers in the somewhat allied case of the Stark effect.
Abstract 1. Indications of the occurrence of fractional quantum numbers have already been found by Kratzer from a study of certain band spectra, and by Curtis in his experimental investigation of helium bands. Heisenberg has also employed half-integral numbers in a quantum theory of the anomalous Zeeman effect with considerable success. The fourth of the slightly extended quantum conditions recently suggested by W. Wilson and applied by the present writer to the Zeeman effect involves a fractional quantum number, as O. W. Richardson has pointed out. K. F. Niessen, in his work on the positively ionised hydrogen molecule, observes certain discontinuities in the energy graphs between symmetrical and asymmetrical models which strongly the occurrence of half-integral quantum orbits for the latter. He refrains, however, from making this assumption, on the ground that it would necessitate the adoption of half numbers in the somewhat allied case of the Stark effect.
Abstract In a previous paper the theory of perturbations was discussed for system which possess only discrete spectra. The theory is now extended to systems with both discrete and continuous spectra, and in addition a slightly more general method of perturbation is considered. The Hamiltonian H of the perturbed systems is split up into three parts H0, H1, H2 (t), two of which can be chosen arbitrarily. The solutions of the perturbed problem can then be worked out in terms of the characteristic functions corresponding to the Hamiltonian H0 + H2 (T), where T is any arbitrary fixed time. In general, however, the series of perturbations does not converge, and different modes of splitting up the Hamiltonian will lead to different formal results. The perturbation theory does usually give the correct physical results, and it is therefore necessary to give some explanation of how this happens. In (I) the asymptotic nature of certain solutions was emphasised, but this does not extend to the more general systems treated here. It is suggested that the usual boundary conditions do not give an adequate description of any but the simplest atomic problems, and that more detailed restrictions, determined by the experimental conditions, ought to be substituted. As this is impossible in practice, an alternative method is to regard the motion as a perturbation on that Hamiltonian H0 + H2 (T) which most nearly corresponds to the experimental conditions. (For example, we may regard an electron as free or bound according as the experiment determines the number of free or bound electrons respectively, whereas in the usual theory the initial Hamiltonian is taken as fundamental.) This criterion, together with the initial conditions, suffices to fix the division of H into the three parts H0, H1, H2 (t) which at first were chosen arbitrarily. The formal solution can be worked out on this basis, and it is then assumed that this solution is asymptotic to the one which would be obtained if the full boundary conditions were used. This suggestion is in accordance with the fact that a hydrogen atom in an electric field gives rise not only to the usual Stark effect, but also to an ionisation effect. In § 2 the perturbation equation are derived in a slightly more general form than has previously been given. The theories of Born,* Dirac and Oppenheimer are particular cases of that given here, and are obtained by particular choices of H0, H1, H2 (t). To obtain Born’s theory we take H1 = 0, and for Oppenheimer’s we take H2 (t) = H2 (T) and T = 0. In Dirac’s theory we have H1 = 0, H2 (0) = 0 and T = 0. Each of these theories is appropriate for the solution of special problems, but the more general method of this paper is required for the full discussion of the perturbation theory.
Abstract 1. The chief object of the present paper is to present a simple system of equations which are competent to determine the frequencies and intensities of the lines in the standard Zeeman effect. By the standard Zeeman effect is meant the type where the terms are given by Landé’s "g"and “γ” formulæ, and where the multiplicity of the two sets of terms is the same. It will probably be held that the theory of such multiplets has been fairly completely understood for the last two years owing to the works of Sommerfeld, Heisenberg, Landé and Pauli, and the main new contribution of the present work is that, whereas these writers gave formulæ valid only either in weak or strong field (except for doublets, where Heisenberg and Jordan give the value for any strength), here we have complete formulæ from which the intensity of any component in any strength of field can be obtained merely from the solution of rather simple algebraic equations. The proper attack on this problem would undoubtedly be by way of the recent work of Heisenberg on the helium spectrum, and his still more recent work on complex spectra; but this theory is still in the making, so that it has not been practicable to apply it here, and to this extent our results are unverified. Their validity rests firstly on a complete verification of all known facts connected with both weak and strong fields (and intermediate fields for doublets), and also on a conformity with the general features of wave mechanics. The work of Heisenberg and Jordan could readily have been adapted to give all the results of the present paper; but it would have been harder to follow because the matrix methods are not so easy for most readers as are spherical harmonics.
The Bohr formulation—the original one—is now largely a matter of history. The usual career of a theory was its fate. It explained much, but it failed at the first signs of complication, even for the simplest molecules as hydrogen and helium. Some results were either only approximate in a loose or incomplete fashion, e.g. in the application of the correspondence principle to intensities, the only reliable predictions being only for an absence of certain lines, or they quite disagreed with experiment, e.g. in the case of the hydrogen molecule and the hydrogen molecule ion, the normal and excited states of helium, the anomalous Zeeman effect, the Ramsauer effect (the phenomenon of the deflection of electrons at low speeds being less than at higher speeds in atomic collisions), and in the treatment of problems where more than one electron was involved. In the last mentioned cases the perturbation method, so successful in solving “many-body” problems in celestial mechanics, failed completely because systems of several electrons, possessing frequencies of the order of light waves, cannot interact classically. A bit of much forced pragmatism was practiced in treating the many electron atom as a variation of the hydrogen atom. Besides, there were technical oddities, like ½ quantum numbers, and impenetrated anologies between visible and X-ray spectra.
Abstract The theory of perturbations of electronic systems has been considered by various writers, and the method is the basis of the solution of most quantum problems. It therefore seems desirable that the theory should be placed on a firm footing, and this necessitates the discussion of the convergence of the series of perturbations, which up to the present has been avoided. This investigation is all the more necessary since the wave equation of the Stark Effect has no solution of the form assumed by Schrödinger in his original discussion of the perturbation theory. Dirac’s theory leads to a system of linear differential equations in an infinite number of variables, and 2 is devoted to the discussion of the existence theorem for such a system by a matrix method. In 3 the application is made to the perturbation theory in the case of perturbations satisfying the conditions assumed in 2. In 4 the theory is extended so as to include more general types of disturbances. It is shown that, though the series of perturbations does not in general converge, yet it usually possesses the same asymptotic character as in the classical theory, and its use can therefore be justified.
Abstract The aim of this paper is to put forward a theory of the simple Zeeman effect which possesses the same general features as those of the corresponding theory in the case of the Stark effect already developed by Epstein and Schwarzschild. These general features may be briefly described as follows:— (1) The steady states of the atom are governed by classical dynamics subject to certain conditions, the quantum restrictions, which define the atom both in the absence and the 'presence of the field; (2) radiation occurs during the period of transition from one steady state to another according to Bohr’s energy relation hv = Wm-Wn.
A theory of the Stark effect based on Schroedinger's ideas is presented. (1) Positions of lines practically coincide with those obtained in the writer's old theory which gave an excellent agreement with experiment. (2) Intensity expressions are obtained in a simple closed form: (a) Components which, in the old theory, had to be ruled out by a special postulate now drop out automatically. (b) The computed intensities of the remaining components check the observed within the limits of experimental error.