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American Journal of Physics

Particle velocity = group velocity: A common assumption in the different theories of Louis de Broglie and Erwin Schrödinger

Nathan Willig Lima, Ricardo Karam

Published Apr 21, 2021• 2 citations
Publisher

Abstract

The identification of particle velocity with the velocity of a wave group was a crucial assumption that resolved important inconsistencies in the theoretical developments of Louis de Broglie and Erwin Schrödinger. Interestingly, this was one of the few common aspects of their work. In this paper, we present a reconstruction of how group velocity became essential for both de Broglie and Schrödinger, as well as a comparison of important differences in their theories. Pedagogical lessons to be extracted from this episode are also outlined.

DOI
10.1119/10.0003165

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The identification of particle velocity with the velocity of a wave group was a crucial assumption that resolved important inconsistencies in the theoretical developments of Louis de Broglie and Erwin Schrödinger. Interestingly, this was one of the few common aspects of their work. In this paper, we present a reconstruction of how group velocity became essential for both de Broglie and Schrödinger, as well as a comparison of important differences in their theories. Pedagogical lessons to be extracted from this episode are also outlined.
Louis de Broglie’s doctoral thesis developed a concept of waves associated with material particles that was soon incorporated into wave mechanics and later supported by experimental demonstrations. De Broglie’s original development, however, relied on an incorrect identification of two quite different relations: the relation between the velocity of a particle and the relation between the group velocity of a wave packet and the velocity of individual waves in the packet. A clarification of this difference and its historial significance is followed by a tentative account of the reasons why de Broglie’s well-known formula proved successful, though the theory supporting it rested on a conceptual confusion. Finally, this development and clarification are related to some of the difficulties involved in the attempts top fashion a coherent interpretation of quantum theory.
Louis-Victor de Broglie secured his place in the history of quantum physics by proposing the existence of matter waves. Following the concept accepted early in the twentieth century that light waves could display properties characteristic of particles (photons), de Broglie suggested the complementary principle that particles could display properties characteristic of waves. Within a few years, this far-reaching proposal was experimentally verified. The well-known formula for the wavelength λ of the wave associated with a particle of momentum p, namely λ=h/p, does not appear in de Broglie’s original paper, which was published in this journal in 1924. What is derived from the starting assumption that the frequency of the wave is the rest energy of the particle divided by Planck’s constant h, is the wave's velocity, which turns out to be c2/v, where v is the particle’s velocity. This velocity, being superluminal, means that the wave cannot transport energy. De Broglie called it a phase wave and postulated a group of such waves having virtually, but not exactly, the same velocity and wavelength. The group velocity of such a large number of almost identical phase waves is then shown, very satisfactorily, to be equal to the velocity of the particle. The proposal by de Broglie inspired Schrödinger to develop the mathematical foundations of wave mechanics. Whereas in conventional quantum theory, phase waves are considered fictional, de Broglie considered them to have a certain physical reality, renaming them pilot or guiding waves – a concept developed later by Bohm in 1952.
In 1924, a young PhD student submitted a thesis containing the germ of an idea that would revolutionize our understanding of the material world. The student was Louis De Broglie, and for a while, he had been meditating on the wave/particle nature of light. De Broglie’s ideas were greeted rather sceptically at first. His PhD supervisor even asked for a second copy so he could send it to Einstein for his opinion. Einstein gave the work his seal of approval. News of De Broglie’s thesis spread around the physics community from 1924 to 1925. In autumn 1925, Erwin Schrodinger was visiting colleagues in Switzerland and was asked to present a summary of De Broglie’s work. Having studied De Broglie’s thesis in some detail he became an enthusiastic supporter. During the Christmas and New Year of 1925/1926, Schrodinger holidayed in the Alps with his mistress. On his return, he had a functioning wave equation worked out.
The proposed paper calls attention towards the unobserved mathematical and conceptual inadequacies persisting in the wave-particle duality and matter wave’s concepts, given by Louis de Broglie. Matter wave’s frequency and phase velocity expressions, shown to be inappropriate, are the consequences of these inadequate concepts. The rectifications in these concepts are presented through the corrected implementation of analogy between light waves and matter waves and thus modified frequency and phase velocity expressions are introduced. The proposed expressions are free from all the inadequacies and negations, contrary to that confronted by de Broglie’s proposed expressions. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. A novel General Quantum Mechanical Wave Equation is proposed involving the modified phase velocity expression, which itself can precisely derive out Schrodinger’s and Dirac’s Equation.
Louis De Broglie is one of the principal architects of quantum theory. It is to him that we owe the idea of the particle as having some sort of wavelike functionality and the wave particle duality. Despite having invented this idea, de Broglie had his doubts and spent much of the rest of his life trying in vain to justify his assertion for which he had won the Nobel Prize. Essentially de Broglie’s notion of the particle as a wave is a restatement of the assumption adopted by Niels Bohr in his model for the hydrogen atom, notably that angular momentum is quantised. De Broglie does however provide us with two useful insights: that the energy levels of the atom are in some way associated with a harmonic sequence and that there is some sort of frequency multiplication process taking place within the atom.