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American Journal of Physics

Frame dependence of the phase of de Broglie waves

J. W. G. Wignall

Published May 1, 1989• 0 citations
Publisher

Abstract

The fact that the phase of de Broglie waves is not invariant under a change of frame described by a Galilean transformation has sometimes been used as an argument against their existence as physically real waves. It is shown, however, that this phase change is an artifact of the Galilean transformation, and does not occur if one uses instead the low-velocity limit of the Lorentz transformation consistently to order (velocity/c)2.

DOI
10.1119/1.16012

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The fact that the phase of de Broglie waves is not invariant under a change of frame described by a Galilean transformation has sometimes been used as an argument against their existence as physically real waves. It is shown, however, that this phase change is an artifact of the Galilean transformation, and does not occur if one uses instead the low-velocity limit of the Lorentz transformation consistently to order (velocity/c)2.
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.
In his famous doctoral dissertation, de Broglie assumed that a massive particle is surrounded in its rest frame by a standing wave. He argued that as observed from another inertial frame this wave becomes the superluminal wave now known as the de Broglie wave. It is shown here that under a Lorentz transformation, such a standing wave becomes, not the de Broglie wave, but a modulated wave moving at the velocity of the particle. It is the modulation that has the superluminal velocity of the de Broglie wave and should be recognized as the true de Broglie wave. De Broglie's demonstrations relied, variously, on his "theorem of the harmony of phases", on a mechanical model, and on a spacetime diagram. It is shown that in each case the underlying wave was inadvertently suppressed. Identified as a modulation, the de Broglie wave acquires a physically reasonable ontology, avoiding the awkward device of recovering the particle velocity from a superposition of such waves. The deeper wave structure implied by this de Broglie wave must also impinge on such issues in quantum mechanics as the meaning of the wave function and the nature of wave-particle duality.
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.
Abstract The forgotten history of de Broglie waves as themselves artifacts of a Lorentz transform, not physical lengths and frequencies to be transformed, causes confusion for students and others. In this paper the de Broglie wavelength is derived and dependence of de Broglie frequency on velocity explained in terms of Einstein synchronized reference frame measurements of a single clock ( 2-on-1 ) versus measurements of an Einstein reference frame by an observer with a single clock ( 1-on-2 ). A new memory aid for the relativity of simultaneity, ‘leading clocks lag,’ is introduced that puts it on a par with the heuristics ‘time dilation’ and ‘length contraction’ for quick and easy problem visualization.
It is argued that the de Broglie wave is not the wave usually supposed, but the relativistically induced modulation of an underlying carrier wave that moves with the velocity of the particle. In the rest frame of the particle this underlying structure has the form of a standing wave. De Broglie also assumed the existence of this antecedent standing wave, but it would appear that he failed to notice its survival as a carrier wave in the Lorentz transformed wave structure. Identified as a modulation, the de Broglie wave acquires a physically reasonable ontology, evidencing a more natural unity between matter and radiation than might otherwise be contemplated, and avoiding the necessity of recovering the particle velocity from a superposition of such waves. Because the Schrödinger and other wave equations for massive particles were conceived as equations for the de Broglie wave, this interpretation of the wave is also relevant to such issues in quantum mechanics as the meaning of the wave function, the nature of wave-particle duality, and the possibility of well-defined particle trajectories.
De Broglie waves were originally derived from the Lorentz Transformation of a standing wave, e−iωte^{-i \omega t}e−iωt, that has no space dependence. It is shown here that a suitable, physically reasonable, standing wave can be constructed from physical waves that propagate at c, subject to the condition that any field line of the wave vector exists on the surface of a sphere at rest in the comoving frame. This result contradicts the classical picture of a point particle emitting a far field that propagates radially away from it, and it is argued that, while the present construction of de Broglie waves is both local and realistic, Bell Inequalities cannot be derived in de Broglie's context.