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Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character

The operational wave equation and the energy levels of the hydrogen atom

George Frederick James Temple

Published May 7, 1930• 11 citations
Publisher

Abstract

Abstract The operational wave equation used in this paper is a generalisation of the linear wave equation introduced by Dirac. The generalisation consists in replacing Dirac’s matrices by any four linear operators (A1, A2, A3, A4), which satisfy Dirac’s conditions, ½ (AmAn + AnAm) = 0 if m ≠ n, (1.1) 1 if m = n. The operational wave equation for free space is therefore.

DOI
10.1098/rspa.1930.0063

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In three earlier papers the author has developed the theory of the operational wave equation, which was originally suggested by Professor Eddington as the most extensive generalisation possible of the linear wave equation devised by Dirac. The wave function,ψ, plays a very minor rô1e in the development of this theory and, in reality, it is introduced simply to provide an operand which shall be patient of the action of the wave operators, A1, A2, A3, A4. The object of this paper is to show that the wave function may be entirely eliminated from the theory, which then takes the form of a “ matrix mechanics,” i.e. a set of relations between matrices representing the coordinates, the momenta and the spin operators.

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Journal of Modern Physics•Jan 1, 2020•2 citations
Abstract The operational wave equation used in this paper is a generalisation of the linear wave equation introduced by Dirac. The generalisation consists in replacing Dirac’s matrices by any four linear operators (A1, A2, A3, A4), which satisfy Dirac’s conditions, ½ (AmAn + AnAm) = 0 if m ≠ n, (1.1) 1 if m = n. The operational wave equation for free space is therefore.
In three earlier papers the author has developed the theory of the operational wave equation, which was originally suggested by Professor Eddington as the most extensive generalisation possible of the linear wave equation devised by Dirac. The wave function,ψ, plays a very minor rô1e in the development of this theory and, in reality, it is introduced simply to provide an operand which shall be patient of the action of the wave operators, A1, A2, A3, A4. The object of this paper is to show that the wave function may be entirely eliminated from the theory, which then takes the form of a “ matrix mechanics,” i.e. a set of relations between matrices representing the coordinates, the momenta and the spin operators.
In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.
Abstract 1. In a recent paper Dirac* has brilliantly removed the defects before existing in the mechanics of the electron, and has shown how the phenomena usually called the “ spinning electron ” fit into place in the complete theory. He applies to he problem the method of q-numbers and, using non-commutative algebra, exhibits the properties of a free electron, and of an electron in a central field of electric force. In a second paper † he also discusses the rules of combination and the Zeeman effect. There are probably readers who will share the present writer’s feeling that the methods of non-commutative algebra are harder to follow, and certainly much more difficult to invent, than are operations of types long familiar to analysis. Wherever it is possible to do so, it is surely better to present the theory in a mathematical form that dates from the time of Laplace and Legendre, if only because the details of the calculus have been so much more thoroughly explored. So the object of the present work is to take Dirac’s system and treat it by the ordinary methods of wave calculus. The chief point of interest is perhaps the solution of the problem of the central field, which can be carried out exactly and leads to Sommerfeld’s original formula for the hydrogen levels. But it is also of some interest to exhibit the relationship of the new theory to the previous equations which were derived empirically by the present writer.* It appears that those equations were an approximation to the new ones, derived by an approximate elimination of two of Dirac’s four wave functions. We shall also review a few other points connected with the free electron, the emission of radiation from an atom and its magnetic moment, and shall outline a discussion of the Zeeman effect.