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Mathematical Proceedings of the Cambridge Philosophical Society

The Two Centre Problem in Wave Mechanics

W. G. Baber, H. R. Hassé

Published Oct 1, 1935• 168 citations
Publisher

Abstract

This problem, in the particular case of the hydrogen molecular ion, is so well known that a detailed reference to the large number of papers dealing with it is unnecessary. The investigations to which chief reference is made will be found at the end of this paper.
DOI
10.1017/s0305004100013566

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In view of the recent experimental determination of the dispersion by atomic hydrogen it seems interesting to apply the theory of dispersion developed by Schrodinger to this case. In this paper we restrict ourselves to an approximation in which terms of the order of relativistic correction are neglected. For this purpose it is simpler to obtain our wave equation by the operational method of Schrodinger and Eckart, as extended by Epstein, for in this way we immediately obtain an equation free of relativistic terms. In what follows, in order to preserve a continuity of the discussion, details of calculations are omitted from the main text and are given as Supplementary Notes at the end of the paper. References will be found immediately preceding the Supplementary Notes.
Approximate expressions are obtained, when the distance R between the nuclei is very large, for that portion of the wave function in the two-centre problem which depends on the hyperbolic coordinate μ. From these expressions the number and the approximate position of the nodes in μ can be deduced and hence the rules can be found by which that state of the combined atom at R = 0 can be determined which corresponds to a given state of the atom when the nuclei are completely separated. These rules are also applicable to the cases where the two atoms which can be formed when the nuclei are completely separated have the same energy. The converse problem of finding what state of the completely separated atom corresponds to a given state of the combined atom at R = 0 can also be solved by the use of the rules.