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Proceedings of the National Academy of Sciences

The Dispersion by Hydrogen-Like Atoms in Undulatory Mechanics

Boris Podolsky

Published Mar 1, 1928• 71 citations
Publisher

Abstract

In view of the recent experimental determination of the dispersion by atomic hydrogen [1] it seems interesting to apply the theory of dispersion developed by Schrödinger [2] 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 Schrödinger [3] and Eckart [4], as extended by Epstein [5], for in this way we immediately obtain an equation free of relativistic terms.

DOI
10.1073/pnas.14.3.253

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In view of the recent experimental determination of the dispersion by atomic hydrogen [1] it seems interesting to apply the theory of dispersion developed by Schrödinger [2] 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 Schrödinger [3] and Eckart [4], as extended by Epstein [5], for in this way we immediately obtain an equation free of relativistic terms.
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.
The quantum mechanical problem of the hydrogen atom is treated by use of a finite difference equation in place of Schrödinger's differential equation. The exact solution leads to a wave vector energy expression that is readily converted to the Bohr-Rydberg formula. (The calculations here reported are limited to spherically symmetric states.) The wave vectors reduce to the familiar solutions of Schrödinger's equation as c --> infinity. The internal consistency and limiting behavior provide support for the view that the equations employed could well constitute an approach to a relativistic formulation of wave mechanics.
The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.