Abstract In 1925, de Broglie hypothesized that any material particle has an associated wave with /h p . Electron diffraction seems to support that Hypothesis. But then, the electron at rest will have infinite wavelength, and infinite wave phase velocity. This says that for a material particle, the de Broglie relation does not hold. Failed attempts to save the postulate, kept the flawed relation, and modified the waves into train waves, pilot waves, probability waves, … to name a few. We keep the de Broglie waves unchanged, and modify the relation. First, we observe that the Planck energy E h used by de Broglie defines virtual electromagnetic waves. Consequently, for any particle, the virtual electromagnetic wavelength is / /c h mc , and / /h m c . Refinement of de Broglie argument, indicates that /h mmay be x , Heisenberg’s uncertainty in the particle location. de Broglie’s later analysis supports this interpretation, and we offer an explanation to particle diffraction as a consequence of Heisenberg’s uncertainty. We apply / /h m c to obtain the dispersion relations for the de Broglie virtual waves.
The validity of de Broglie's equation is checked experimentally for electrons of 24 to 64 electron-kilovolts energy. The experiment consists essentially of deflecting a beam of cathode rays by an electrostatic field and, keeping the deflection constant, determining the relation between the intensity of the deflecting field,
V, and the de Broglie wave-length, $\ensuremath{\lambda}$, of the deflected electrons. On the assumption that the motion of the electrons is in accordance with the laws of relativistic mechanics, and that the de Broglie wave-length is inversely proportional to the momentum, it is shown theoretically that the expression ${\ensuremath{\lambda}}^{2}V{(1+\frac{{h}^{2}}{{{m}_{0}}^{2}{c}^{2}{\ensuremath{\lambda}}^{2}})}^{\frac{1}{2}}$ should be independent of $\ensuremath{\lambda}$. Within the limits of experimental error this expression is found to be independent of $\ensuremath{\lambda}$, thereby demonstrating that the de Broglie equation, $\ensuremath{\lambda}=\frac{h}{\mathrm{mv}}$, combined with the relativity expression for
m, is adequate to describe the facts.
De Broglie’s association of a wave to particles is a fundamental concept in the quantum mechanical description of nature. The wave oscillation is referred to alternatively as the “de Broglie clock”, the “Compton clock”, or the “de Broglie periodic phenomenon”. In the present paper it is shown that Dirac’s relativistic quantum mechanics, complemented with the dynamical time operator recently introduced, provides a consistent theoretical description of: (i) the generation of the de Broglie wave through Lorentz boosts; and (ii) the characteristics of the resonance observed in electron channeling through thin crystals as responding to both the periodicity derived from the adjustment of the de Broglie period to the crystal interatomic distance (resonance energy) and the periodicity of the predicted trembling motion (Zitterbewegung). One can conclude that the channeling experiments provide the first direct evidence of the electron Zitterbewegung, and that the de Broglie period is an intrinsic property of matter arising from a self-adjoint dynamical time operator.