The wave equations of the electron

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.