The wave mechanics shows that the momentum 3-vector of an electron of a rest mass [m.sub.0] (in vacuum) is given by the de Broglie relation
In quantum mechanics, momentum of microparticles p=mv, where m and v are respectively, the rest mass and the speed of the microparticles, with its wavelength [lambda] associated by de Broglie relation
The force difference [DELTA][F.sub.4] = 0 gives the de Broglie relation
The expression m[upsilon][lambda] = h used in (1) to arrive at the total electron kinetic energy is the de Broglie relation expressed in simple, physically intuitive terms: the de Broglie relation yields the product of the electron mass m, its average velocity [upsilon], and the path length [lambda] over which its instantaneous velocity varies.
Thus Beckmann's de Broglie relation is in relativistic agreement with the PV result.
The equation [DELTA][F.sub.3] = 0 from the final two brackets yields the de Broglie relation
567], in order to apply the de Broglie relation, the following assumption is made
From this equation the momentum of the electron is calculated as [absolute value of p] = [??]/c, and from the de Broglie relation it follows that [lambda] = hc/[??].
associates a Compton wavelength [[lambda].sub.c](or a Compton radius [r.sub.c] = [[lambda].sub.c]/2[pi]) with the particle mass m, while the de Broglie relation
The de Broglie relations
derived from the electron and proton coupling to the Planck vacuum state.
This phase wave in terms of transmitting the particle mass at the speed v and angular frequency [[OMEGA].sub.d] = 2[pi]v/[[LAMBDA].sub.d], with [[LAMBDA].sub.d] and [[OMEGA].sub.d] obeying the de Broglie relations, represents a de Broglie wave.
The solutions for a basic material particle generally in motion, with the charge quantity (accompanied with a spin) and energy of the charge as the sole inputs, predict accurately the inertial mass, total wave function, total energy equal to the mass times [c.sup.2], total momentum, kinetic energy and linear momentum of the particle, and that the particle is a de Broglie wave, it obeys Newton's laws of motion, de Broglie relations, Schrodinger equation in small geometries, Newton's law of gravitation, and Galilean-Lorentz-Einstein transformation at high velocities.