We set up the Maxwell's equations and the corresponding classical wave equations for the electromagnetic waves which together with the generating source, a traveling oscillatory charge of zero rest mass, comprise a particle traveling in the force field of an usual conservative potential and an additional frictional force $f$. At the de Broglie wavelength scale and in the classic-velocity limit, the total wave equation decomposes into a component equation describing the particle kinetic motion, which for $f=0$ identifies with the usual linear Schrödinger equation as previously. The $f$-dependent probability density presents generally an observable diffusion current of a real diffusion constant; this and the particle's usual quantum diffusion current as a whole are under adiabatic condition conserved and obey the Fokker-Planck equation. The corresponding extra, $f$-dependent term in the Hamiltonian operator identifies with that obtained by H.-D. Doebner and G.A. Goldin. The friction produces to the particle's wave amplitude a damping that can describe well the effect due to a radiation (de)polarization field, which is always by-produced by the particle's oscillatory charge in a (nonpolar) dielectric medium. The radiation depolarization field in a dielectric vacuum has two separate significances: it participates to exert on another particle an attractive, depolarization radiation force which resembles in overall respects Newton's universal gravity as we showed earlier, and it exerts on the particle itself an attractive, self depolarization radiation force whose time rate gives directly the frictional force $f$.
We set up the Maxwell's equations and the corresponding classical wave equations for the electromagnetic waves which together with the generating source, a traveling oscillatory charge of zero rest mass, comprise a particle traveling in the force field of an usual conservative potential and an additional frictional force f. At the de Broglie wavelength scale and in the classic-velocity limit, the total wave equation decomposes into a component equation describing the particle kinetic motion, which for f=0 identifies with the usual linear Schrödinger equation as previously. The f-dependent probability density presents generally an observable diffusion current of a real diffusion constant; this and the particle's usual quantum diffusion current as a whole are under adiabatic condition conserved and obey the Fokker-Planck equation. The corresponding extra, f-dependent term in the Hamiltonian operator identifies with that obtained by H.-D. Doebner and G.A. Goldin. The friction produces to the particle's wave amplitude a damping that can describe well the effect due to a radiation (de)polarization field, which is always by-produced by the particle's oscillatory charge in a (nonpolar) dielectric medium. The radiation depolarization field in a dielectric vacuum has two separate significances: it participates to exert on another particle an attractive, depolarization radiation force which resembles in overall respects Newton's universal gravity as we showed earlier, and it exerts on the particle itself an attractive, self depolarization radiation force whose time rate gives directly the frictional force f.
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.
We set up the classical wave equation for a particle formed of an oscillatory zero-rest-mass charge together with its resulting electromagnetic waves, traveling in a potential field V in a susceptible vacuum. The waves are Doppler-displaced upon the source motion, and superpose into a total, traveling- and in turn a standing- beat wave, or de Broglie phase wave, described by a corresponding total classical wave equation. By back-substitution of the explicitly known total, standing beat wave function and upon appropriate reductions at classic-velocity limit, we separate out from the total a component wave equation describing the kinetic motion of particle, which is equivalent to the Schrödinger equation. The Schrödinger wave function follows to be the envelope function of the standing beat wave at classic-velocity limit.
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.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
A method previously proposed by the same author, of solving the equations of motion in the presence of friction and an external stochastic force, is applied to the nonrelativistic Dirac equation for a charged bound pointlike particle in a black-body radiation field. It separates the particle velocity field, under the assumtion of stationarity, into a position-dependent component and a randomly fluctuating component depending mainly upon the time. This description of the possible stationary states of the bound particle in the random force field is taken as a starting point for establishing a diffusion equation in configuration space. Since we identify the first component with the drift velocity, we show that it must be the solution to a modified Hamilton–Jacobi equation by using a term that is the counterpart of the quantum modification to the same equation, which has as a first approximation exactly the same form. This term gives results proportional to the diffusion coefficient and, thereby, to the spectral density of the external random force. The diffusion equation that is obtained has complex coefficients and therefore it defines a probability density for the complex values of the variable coordinate. We propose an interpretation of this probability density, based on a specific example.
We have argued in previous notes (1) that the “wave” nature of quantum mechanics follows from the Lorentz invariant for a constant p: A= -Et+px which holds for both a particle with rest mass and a photon. In A, t and x are separated indicating independence which violates the Newtonian view x(t). “A” leads to dA/dx partial = p and dA/dt= -E. Creating eigenvalue equations yields: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt). Thus, x and t may be separated. Given that exp(ipx) exists in all of space (because there is no time), it represents a somewhat spatially invariant probability. In the case of a photon, Maxwell showed one may write wave equations for the electric El and magnetic B fields using his electromagnetic equations with no charge or currents.. He showed the wave travels at the speed of light in a vacuum and thus characterized light as an electromagnetic wave. This is not the same statement as exp(ipx) which is a probability wave. One may note that Maxwell’s equations are Lorentz invariant, so the ideas of the first paragraph should hold. In this note, however, we consider the Poynting equation: d/dt partial (.5( eoElEl+1/uoBB)) + 1/uo grad dot El x B = function(charge and current). 1/uo ElxB is considered to be a momentum density. We note Maxwell’s equations contain d/dt partial and d/dx partial, so time and space are considered independent (unlike x(t)), and act on El and B. The presence of charge and current densities breaks spatial invariance i.e. creates local regions. We argue in this note that d/dx and d/dt are generators of translations in space and time. We further argue that Maxwell’s theoretical creation of a photon follows from introducing spatial invariance into the problem, i.e. removing all charge and current densities. Spatial invariance in Newtonian mechanics, however, is associated with a constant i.e. a constant speed and momentum and equal probability to be at any x. Given that d/dx and d/dt appear, one has a function of space and time which seems to be completely at odds with spatial invariance and the idea of a constant momentum. We have argued in (2), however, that constant motion, even though represented by v=constant, is associated with the force that created the momentum, and this manifests itself in a spatial density which is probabilistic. Thus the presence of d/dt and d/dx in Maxwell’s electromagnetic equations, even when charge and current densities are removed, implies the same thing. Thus we argue that Maxwell’s electromagnetic equations are probabilistic. We investigate these ideas in this note. We further argue that it is spatial invariance, together with the presence of d/dx partial which yields the periodic form exp(ipx). Thus it is not a coincidence that wave equations appear for El and B.