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European Journal of Physics

On the de Broglie waves

Janez Strnad, W Kuhn

Published Jul 1, 1985• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0143-0807/6/3/009

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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 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.
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.
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.
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.
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 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.