The original wave theory proposed by de Broglie necessitates a stationary (localized) monochromatic plane wave of high frequency νo associated with particle. As particle moves the de Broglie wavelength appears as an observable (a relativistic effect). The nature and interpretation of high frequency oscillations remained unrevealed so far. This paper presents a self trapped circular Electromagnetic (EM) travelling wave model of Electron (specifically) and other charged particles generally. Curling EM wave is localized (stationary) and still a plane monochromatic travelling wave, against the customary wave packet approach, where number of frequencies are made to interfere to build a stationary localized wave-packet. The model reveals facts such as interpretation of wave function ψ (de Broglie wave) as a relativistic effect of a real wave function and removes the discrepancy of very high phase velocity (c 2 /v) associated with it. The model also incorporates zitterbewegung of electron and 4π rotation needed, for a complete round of wave cycle.
The original wave theory proposed by de Broglie necessitates a stationary (localized) monochromatic plane wave of high frequency νo associated with particle. As particle moves the de Broglie wavelength appears as an observable (a relativistic effect). The nature and interpretation of high frequency oscillations remained unrevealed so far. This paper presents a self trapped circular Electromagnetic (EM) travelling wave model of Electron (specifically) and other charged particles generally. Curling EM wave is localized (stationary) and still a plane monochromatic travelling wave, against the customary wave packet approach, where number of frequencies are made to interfere to build a stationary localized wave-packet. The model reveals facts such as interpretation of wave function ψ (de Broglie wave) as a relativistic effect of a real wave function and removes the discrepancy of very high phase velocity (c 2 /v) associated with it. The model also incorporates zitterbewegung of electron and 4π rotation needed, for a complete round of wave cycle.
We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c ; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.
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.
Zitterbewegung and de Broglie matter waves are both phenomena associated with the quantum-mechanical properties of the electron, but they are usually discussed in different contexts. Zitterbewegung is understood as a rapid oscillatory motion appearing in the Dirac equation, whereas the de Broglie matter wave is understood as a wave corresponding to the linear momentum of a particle. Some previous studies have attempted to derive the de Broglie matter wave from an internal clock of the electron or from Zitterbewegung. In the present study, however, no causal relation in which the matter wave is generated by Zitterbewegung is assumed. Instead, the two are regarded as different geometrical representations of a common phase degree of freedom. Specifically, Zitterbewegung is interpreted as a phenomenon corresponding to an internal rotational degree of freedom, while the de Broglie matter wave is interpreted as spatial phase progression associated with linear momentum. and are interpreted, respectively, as the rates of phase progression in the temporal and spatial directions. Furthermore, using the relation introduced in previous work, and reinterpreting it as the ratio of the internal rotational velocity to the external propagation velocity, this study shows the possibility of describing internal rotation, translational motion, matter waves, and the fine-structure constant within a common phase geometry.
The electromagnetic component waves, comprising together with their generating oscillatory massless charge a material particle, will be Doppler shifted when the charge hence particle is in motion, with a velocity v, as a mere mechanical consequence of the source motion. We illustrate here that two such component waves generated in opposite directions and propagating at speed c between walls in a one-dimensional box, superpose into a traveling beat wave of wavelength ${\mitΛ}_d$$=(\frac{v}{c}){\mitΛ}$ and phase velocity c2/v+v which resembles directly L. de Broglie's hypothetic phase wave. This phase wave in terms of transporting the particle mass at the speed v and angular frequency ${\mitΩ}_d=2πv /{\mitΛ}_d$, with ${\mitΛ}_d$ and ${\mitΩ}_d$ obeying the de Broglie relations, represents a de Broglie wave. The standing-wave function of the de Broglie (phase) wave and its variables for particle dynamics in small geometries are equivalent to the eigen-state solutions to Schrödinger equation of an identical system.
Abstract We present a classical hydrodynamic analog of free relativistic quantum particles inspired by de Broglie's pilot wave theory and recent developments in hydrodynamic quantum analogs. The proposed model couples a periodically forced Klein-Gordon equation with a nonrelativistic particle dynamic equation. The coupled equations may represent both quantum particles and classical particles driven by the gradients of locally excited Faraday waves. Exact stationary solutions of the coupled system reveal a highly nonlinear mechanism responsible for the self-propulsion of free particles, leading to the onset of unsteady motion. Although the model is essentially nonrelativistic, a stabilizing mechanism for any particle traveling close to the wave signaling speed emerges through the coupling with the wavefield. Consequently, inline particle oscillations comparable to de Broglie's wavelength formula are realized through this fully-classical model, suggesting a new classical interpretation for the motion of relativistic quantum particles.
De Broglie waves are a simple consequence of special relativity applied to the complex-phase oscillations of stationary states. As de Broglie showed in his doctoral thesis, the synchronized oscillations of an extended system at rest, even a classical one, become de Broglie-like waves when boosted to finite velocity. The waves illustrate the well-known but seldom demonstrated relativistic effect of clock desynchroniation (or dephasing) in moving frames. Although common manifestations of stationary-state oscillations in interference experiments are sensitive only to energy differences, de Broglie wavelengths are inversely proportional to rest-frame oscillation frequency, and their observed values require that the oscillation frequencies are proportional to the the total absolute energy, including the rest component mc2. PACS Nos.: 03.65.Ta, 03.30.+p, 01.65.+g