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American Journal of Physics

De Broglie’s relativistic phase waves and wave groups

Harvey R. Brown, Roberto de Andrade Martins

Published Dec 1, 1984• 10 citations
Publisher

Abstract

Louis de Broglie’s original theory of matter waves has until very recently received relatively little treatment in the literature on history and foundations of physics. This paper attempts to clarify a number of aspects of de Broglie’s theory, as formulated primarily in his famous thesis, which still give rise to confusion or debate. To this end, a re-examination is made of the scope and validity of the relativistic physics employed in de Broglie’s arguments concerning the existence of phase waves and their physical implications. Recent doubts concerning the soundness of de Broglie’s introduction of a wave group associated with the quantum are also analyzed in detail, and it is argued that the wave group indeed has a problematic, if minor, role in the overall theory.

DOI
10.1119/1.13743

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Louis de Broglie’s original theory of matter waves has until very recently received relatively little treatment in the literature on history and foundations of physics. This paper attempts to clarify a number of aspects of de Broglie’s theory, as formulated primarily in his famous thesis, which still give rise to confusion or debate. To this end, a re-examination is made of the scope and validity of the relativistic physics employed in de Broglie’s arguments concerning the existence of phase waves and their physical implications. Recent doubts concerning the soundness of de Broglie’s introduction of a wave group associated with the quantum are also analyzed in detail, and it is argued that the wave group indeed has a problematic, if minor, role in the overall theory.
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.
On the occasion of the sixtieth anniversary of Louis de Broglie's thesis, the concept of phase waves, is re-examined. After a brief historical survey a modern conceptual presentation of these waves is described via phase invariance. In this respect de Broglie's hypothesis appears not to be self-consistent. Phase waves in galilean relativity are briefly discussed. As an interesting pedagogical device, inverse phase waves in two-dimensional special relativity are presented. Finally, some remarks are added concerning the development of concepts and theories.
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.
An account of de Broglie’s introduction of phase waves is given that emphasizes that they are a relativistic concept. The properties of the frequency, wavelength, and velocity of de Broglie waves are found under a Lorentz transformation. A nonrelativistic specialization of phase waves is obtained, and the appropriate frequency, wavelength, and velocity are defined. The transformation properties of these parameters under a Galilean transformation are given. The concept of a group velocity for phase waves is re-examined and shown to be applicable.
This article proposes a conceptual analysis of the reasoning developed by Louis de Broglie in his 1924 thesis Recherches sur la théorie des quanta, particularly the pages devoted to the derivation of the phase wave. By distinguishing the proper frequency associated with the particle at rest, the frequency observed from the stationary frame, and the frequency derived from the relativistic energy of the corpuscle, we show that these different frequencies do not appear to possess the same physical status in de Broglie’s original reasoning. We therefore examine the hypothesis according to which the introduction of the phase wave primarily responds to the necessity of maintaining relativistic coherence between two distinct interpretations of frequency. This study does not challenge the physical validity of the de Broglie relation linking momentum and wavelength, but rather proposes a critical re-examination of the conceptual status of the frequencies involved in the original 1924 construction. Finally, we discuss the possibility of future geometrical or topological reformulations of this relation without explicit recourse to a superluminal hase wave.
The de Broglie relation is revisited in connection with an ab initio relativistic description of particles and waves, which is the same treatment that historically led to this famous relation. In the same context of the Minkowski four-vector formalism, we also discuss the phase and the group velocity of a matter wave, explicitly showing that both transform as ordinary velocities under a Lorentz transformation. We show that such a transformation rule is a necessary condition for the covariance of the de Broglie relation, and stress the pedagogical value of the Einstein-Minkowski-Lorentz relativistic context in the presentation of the de Broglie relation.