On Schrödinger's equation, Hertz's mechanics and Van Vleck's determinant

Abstract

There has been much research on Schrödinger's route to what we now call Schrödinger's equation. Various authors disagree as to the exact nature of the influence of each of the physicists he cites—and of some that he does not. This paper, intended for graduate students of and researchers in quantum theory, clarifies Schrödinger's original aims in formulating a wave equation for matter, discusses how far he fulfilled his original aspirations and in what respects he fell short of his goal. An analysis of Schrödinger's foundational paper enables us to distinguish between a formal and an epistemological part, and consider the input of the physicists cited on the basis of the part in which each reference occurs. It turns out, for instance, that Hamilton's optical-mechanical analogy belongs entirely to the epistemological part. Indeed, no element of this analogy plays any role in the formal part of Schrödinger's argument. Instead of basing his theory on this analogy, as is often done nowadays in the physics literature and even in the history of science, we maintain that the aim of Schrödinger's project was to represent wave phenomena by a ‘wave’ in configuration- or q-space, paralleling Hertz's treatment in his Mechanics (1894). The influence of this book on Schrödinger's foundational paper is demonstrated by an analysis of his unpublished paper: Hertz's Mechanics and Einstein's Theory of Gravitation. This approach enables us to dispense with the optical-mechanical analogy in tracing the route to Schrödinger's equation. We also discuss the curious role of the Van Vleck determinant as the ‘missing link’ in taking the classical limit of Schrödinger's wavefunction. A concluding section discusses the relation of some of Schrödinger's earlier and later work to the development of quantum field theory.