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Constructing de Broglie's Periodic Phenomena

Andrew Laidlaw

Published Feb 1, 2016• 0 citations
OpenAlex

Abstract

De Broglie waves were originally derived from the Lorentz Transformation of a standing wave, $e^{-i \omega t}$, 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.

Related papers

The de Broglie Wave as Evidence of a Deeper Wave Structure

Daniel Shanahan

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.

arXiv:1503.02534•Mar 6, 2015•0 citations

The Other de Broglie Wave

Daniel Shanahan

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.

•Mar 6, 2015•1 citation

About the Nature of De Broglie Wave

Isaak Man’kin

In this article, it is shown that if a particle is at rest and has some harmonic oscillation, then a traveling particle with a constant speed corresponds to a De Broglie Wave.In 1924, De Broglie proposed that particles of matter exhibit wave like behavior, the expression for which became fundamental in the creation of wave mechanics (quantum mechanics).The main equation for this is the Schrödinger equation, which in the case of free motion of a particle becomes the expression for the De Broglie wave.This hypothesis is supported by many early experiments on the diffraction of electrons, and later the development of modern physics.However until now, the origin of this hypothesis remains unclear.We show that the expression for the De Broglie wavelength can be obtained from a more general well-known concept, that any particle at rest has a corresponding harmonic oscillation that transforms into a wave of a moving particle with the help of relativistic change of coordinates (the Lorentz transformations).In this way, a De Broglie wave can be explained as a spacetime wave.

OALib•Jan 1, 2022•1 citation

Frame dependence of the phase of de Broglie waves

J. W. G. Wignall

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.

American Journal of Physics•May 1, 1989•0 citations

Clock, Boost, Wave: A Kinematic Origin of the de Broglie Relation

Ping Zhang

The de~Broglie relation $\lambda = h/p$ is among the most well-confirmed results in modern physics, yet the kinematic origin of the associated matter wave remains conceptually obscure. This paper presents a kinematic analysis leading to the de~Broglie relation. The only assumption is the existence of a globally defined phase field with uniform rest-frame oscillation, a structure known to occur in superfluids and Bose--Einstein condensates. Under a Lorentz boost, the scalar transformation of the phase field converts temporal periodicity directly into spatial periodicity, yielding a wave number $k' = \gamma\omega_0 v/c_s^2$ and a phase velocity $v_{\mathrm{phase}} = c_s^2/v$. No wave equation, dynamical model, or quantization procedure is required. The result follows solely from the relativity of simultaneity applied to a scalar phase field. When the characteristic frequency and invariant speed are identified as $\omega_0 = mc^2/\hbar$ and $c_s = c$, the resulting wavelength and phase velocity reproduce the standard de~Broglie relations. The logical architecture inverts that of de~Broglie's original construction: rather than postulating a separate clock and wave linked by phase harmony, the wave emerges as the Lorentz-transformed image of a single coherent phase clock. The phase harmony that de~Broglie introduced as an independent postulate becomes a consequence of Lorentz covariance. The analysis establishes a precise kinematic correspondence without asserting that the physical vacuum is a coherent medium. Whether this correspondence reflects a deeper physical principle or a mathematical analogy remains an open question.

•Jun 1, 2026•0 citations

Analogue Gravity and the de Broglie wave: a Missed Opportunity

Shanahan, Daniel

In a small book entitled Ondes et Mouvements, published in February 1926, Louis de Broglie described the wave, now known as the de Broglie wave, as a modulation or beating effect of undulatory form induced in the structure of the particle by the failure of simultaneity. Considered in this way, the de Broglie wave is neither ontologically distinct, nor in any way separate, from the particle, but like the Fitzgerald-Lorentz contraction is a distortion in the structure of the particle itself. So understood, the de Broglie wave is a physically real phenomenon, capable of describing for the particle, a well-defined and physically realistic trajectory. In comparison, and as I argue in this paper, the wave functions that emerge as solutions to the Schrödinger and Klein-Gordon equations are better regarded as mathematical constructs, albeit constructs of significant utility, identifying the wave number and frequency that the particle would have at each point of space if it were in fact at that point of space. A particular concern of this paper will be to show that the de Broglie wave would emerge as such a distortion of structure in certain sonic quasiparticles proposed in the context of Analogue Gravity for the purpose of simulating the Lorentz transformation.

•Sep 16, 2024•0 citations
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.
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.
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.
In this article, it is shown that if a particle is at rest and has some harmonic oscillation, then a traveling particle with a constant speed corresponds to a De Broglie Wave.In 1924, De Broglie proposed that particles of matter exhibit wave like behavior, the expression for which became fundamental in the creation of wave mechanics (quantum mechanics).The main equation for this is the Schrödinger equation, which in the case of free motion of a particle becomes the expression for the De Broglie wave.This hypothesis is supported by many early experiments on the diffraction of electrons, and later the development of modern physics.However until now, the origin of this hypothesis remains unclear.We show that the expression for the De Broglie wavelength can be obtained from a more general well-known concept, that any particle at rest has a corresponding harmonic oscillation that transforms into a wave of a moving particle with the help of relativistic change of coordinates (the Lorentz transformations).In this way, a De Broglie wave can be explained as a spacetime wave.
The de~Broglie relation λ=h/p\lambda = h/pλ=h/p is among the most well-confirmed results in modern physics, yet the kinematic origin of the associated matter wave remains conceptually obscure. This paper presents a kinematic analysis leading to the de~Broglie relation. The only assumption is the existence of a globally defined phase field with uniform rest-frame oscillation, a structure known to occur in superfluids and Bose--Einstein condensates. Under a Lorentz boost, the scalar transformation of the phase field converts temporal periodicity directly into spatial periodicity, yielding a wave number k′=γω0v/cs2k' = \gamma\omega_0 v/c_s^2k′=γω0​v/cs2​ and a phase velocity vphase=cs2/vv_{\mathrm{phase}} = c_s^2/vvphase​=cs2​/v. No wave equation, dynamical model, or quantization procedure is required. The result follows solely from the relativity of simultaneity applied to a scalar phase field. When the characteristic frequency and invariant speed are identified as ω0=mc2/ℏ\omega_0 = mc^2/\hbarω0​=mc2/ℏ and cs=cc_s = ccs​=c, the resulting wavelength and phase velocity reproduce the standard de~Broglie relations. The logical architecture inverts that of de~Broglie's original construction: rather than postulating a separate clock and wave linked by phase harmony, the wave emerges as the Lorentz-transformed image of a single coherent phase clock. The phase harmony that de~Broglie introduced as an independent postulate becomes a consequence of Lorentz covariance. The analysis establishes a precise kinematic correspondence without asserting that the physical vacuum is a coherent medium. Whether this correspondence reflects a deeper physical principle or a mathematical analogy remains an open question.
In a small book entitled Ondes et Mouvements, published in February 1926, Louis de Broglie described the wave, now known as the de Broglie wave, as a modulation or beating effect of undulatory form induced in the structure of the particle by the failure of simultaneity. Considered in this way, the de Broglie wave is neither ontologically distinct, nor in any way separate, from the particle, but like the Fitzgerald-Lorentz contraction is a distortion in the structure of the particle itself. So understood, the de Broglie wave is a physically real phenomenon, capable of describing for the particle, a well-defined and physically realistic trajectory. In comparison, and as I argue in this paper, the wave functions that emerge as solutions to the Schrödinger and Klein-Gordon equations are better regarded as mathematical constructs, albeit constructs of significant utility, identifying the wave number and frequency that the particle would have at each point of space if it were in fact at that point of space. A particular concern of this paper will be to show that the de Broglie wave would emerge as such a distortion of structure in certain sonic quasiparticles proposed in the context of Analogue Gravity for the purpose of simulating the Lorentz transformation.