Developing de Broglie Wave

Abstract

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 ${\mit\Lambda}_d$$=(\frac{v}{c}){\mit\Lambda}$ and phase velocity $c^2/v+v$ 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 ${\mit\Omega}_d=2\pi v /{\mit\Lambda}_d$, with ${\mit\Lambda}_d$ and ${\mit\Omega}_d$ 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 Schr\"odinger equation of an identical system.