Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:physics/0411134

Inference of Schrödinger's Equation from Classical-Mechanical Solution

J. X. Zheng-Johansson, Johansson, P-I.

Published Nov 15, 2004• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.physics/0411134

Related papers

Developing de Broglie Wave

J. X. Zheng-Johansson, Johansson, P-I

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 $c^2/v+v$ 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 ${\mitΩ}_d=2πv /{\mitΛ}_d$, 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.

arXiv:physics/0608265•Aug 27, 2006•5 citations

Inference of Schrodinger's equation from classical wave mechanics (1) single charge in zero potential field

J. X. Zheng-Johansson, Pär I. Johansson

We show that the classical wave equation for a single oscillatory charge which may be in general also traveling in a potential field $V$, embraces a separable component equation describing the particle dynamics. We show the latter is equivalent to Schr\\"odinger's wave equation. In this first paper of a series we infer Schr\\"odinger's wave equation for such a charged system in zero potential field.

•Nov 15, 2004•0 citations

Dirac equation for electrodynamic model particles

J. X. Zheng-Johansson

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.

Journal of Physics Conference Series•Aug 1, 2008•5 citations

A unification scheme for classical and quantum mechanics at all velocities

John Johansson, Johansson, P-I.

From a Newtonian-Maxwellian solution for a perturbed vacuum with a physical structure constructed based on pivotal experimental observations, we have achieved a general scheme for the formation of basic material particles. A basic particle, which may be e.g. an electron, is composed of a tiny free aether-pole (a bare charge) and the mechanical wave disturbances -- identifying with electromagnetic waves -- generated by it in the medium. When in motion, as a result of a first kind source effect, this particle wave exhibits all of wave and dynamic properties known for a de Broglie wave, and is here called a Newton- de Broglie (NdB) particle wave. In a confined space, the Newtonian solution for the NdB particle wave is equivalent to that given by Schrodinger's quantum mechanics. Through this general scheme we have accomplished a basic task of the unification of the classical- and the quantum- mechanics, both in terms of the deduction of the latter from the former, and the convergence of the latter into the former at high velocities. Through completing the task, we unfold the origins of a series of phenomena including the electromagnetic waves, the electromagnetic radiation and absorption, atomic and thermal excitations, the inertial mass, the Schrodinger's wavefunction and de Broglie wave, the Heisenberg's uncertainty relation, the de Broglie relations, the simultaneous existence of electron and positron or generally of particles and their anti-particles, the (rest) mass-energy equivalence relation, etc. The general scheme facilitates also a Theory of Relative Motion which we present in a separate paper, II; a series of followed studies are planned. (An original report of the scheme with a Preface remarking on the changes in later publications.)

arXiv:physics/0412168•APS•Dec 27, 2004•5 citations

The de Broglie Wave as Evidence of a Deeper Wave Structure

Daniel Shanahan

It is argued that the de Broglie wave is not the wave usually supposed, but the relativistically induced modulation of an underlying carrier wave that moves with the velocity of the particle. In the rest frame of the particle this underlying structure has the form of a standing wave. De Broglie also assumed the existence of this antecedent standing wave, but it would appear that he failed to notice its survival as a carrier wave in the Lorentz transformed wave structure. Identified as a modulation, the de Broglie wave acquires a physically reasonable ontology, evidencing a more natural unity between matter and radiation than might otherwise be contemplated, and avoiding the necessity of recovering the particle velocity from a superposition of such waves. Because the Schrödinger and other wave equations for massive particles were conceived as equations for the de Broglie wave, this interpretation of the wave is also relevant to such issues in quantum mechanics as the meaning of the wave function, the nature of wave-particle duality, and the possibility of well-defined particle trajectories.

arXiv:1503.02534•Mar 6, 2015•0 citations

Does the Schrödinger wave function describe a real physical wave?

Kurt Jung

In 1924 Louis de Broglie presented the phase wave model [1]. He postulated that particle and wave coexist. Both objects are coupled to each other by the phase harmony principle, which always tends to synchronize the oscillations of wave and particle. The escort wave model [2] replenishes the phase wave model by attributing a stochastic velocity component and a potential dependent rest mass to the particle. Moreover it introduces a novel method to deal with standing waves. In this new model a particle's trajectory is continuous but not deterministic because the particle is not only subjected to regular forces derived from external potentials but also to fluctuating forces. On this condition the space dependence of the wave escorting a particle agrees with the Schrödinger wave function. By a measurement process the phase harmony coupling is broken off. Contrary to the standard interpretation of quantum mechanics the escort wave does not collapse but disappears in the ocean of fluctuating waves, because it is no longer stabilized by the particle. The particle immediately starts to build up a new escort wave by adjusting its phase to the phases of fluctuating waves, whose frequencies and rest systems nearly agree with the corresponding magnitudes of the particle.

Journal of Physics Conference Series•Jul 8, 2011•2 citations
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 VVV 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 vvv, 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 ccc 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+vc^2/v+vc2/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 vvv 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.
We show that the classical wave equation for a single oscillatory charge which may be in general also traveling in a potential field VVV, embraces a separable component equation describing the particle dynamics. We show the latter is equivalent to Schr\\"odinger's wave equation. In this first paper of a series we infer Schr\\"odinger's wave equation for such a charged system in zero potential field.
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.
From a Newtonian-Maxwellian solution for a perturbed vacuum with a physical structure constructed based on pivotal experimental observations, we have achieved a general scheme for the formation of basic material particles. A basic particle, which may be e.g. an electron, is composed of a tiny free aether-pole (a bare charge) and the mechanical wave disturbances -- identifying with electromagnetic waves -- generated by it in the medium. When in motion, as a result of a first kind source effect, this particle wave exhibits all of wave and dynamic properties known for a de Broglie wave, and is here called a Newton- de Broglie (NdB) particle wave. In a confined space, the Newtonian solution for the NdB particle wave is equivalent to that given by Schrodinger's quantum mechanics. Through this general scheme we have accomplished a basic task of the unification of the classical- and the quantum- mechanics, both in terms of the deduction of the latter from the former, and the convergence of the latter into the former at high velocities. Through completing the task, we unfold the origins of a series of phenomena including the electromagnetic waves, the electromagnetic radiation and absorption, atomic and thermal excitations, the inertial mass, the Schrodinger's wavefunction and de Broglie wave, the Heisenberg's uncertainty relation, the de Broglie relations, the simultaneous existence of electron and positron or generally of particles and their anti-particles, the (rest) mass-energy equivalence relation, etc. The general scheme facilitates also a Theory of Relative Motion which we present in a separate paper, II; a series of followed studies are planned. (An original report of the scheme with a Preface remarking on the changes in later publications.)
It is argued that the de Broglie wave is not the wave usually supposed, but the relativistically induced modulation of an underlying carrier wave that moves with the velocity of the particle. In the rest frame of the particle this underlying structure has the form of a standing wave. De Broglie also assumed the existence of this antecedent standing wave, but it would appear that he failed to notice its survival as a carrier wave in the Lorentz transformed wave structure. Identified as a modulation, the de Broglie wave acquires a physically reasonable ontology, evidencing a more natural unity between matter and radiation than might otherwise be contemplated, and avoiding the necessity of recovering the particle velocity from a superposition of such waves. Because the Schrödinger and other wave equations for massive particles were conceived as equations for the de Broglie wave, this interpretation of the wave is also relevant to such issues in quantum mechanics as the meaning of the wave function, the nature of wave-particle duality, and the possibility of well-defined particle trajectories.
In 1924 Louis de Broglie presented the phase wave model [1]. He postulated that particle and wave coexist. Both objects are coupled to each other by the phase harmony principle, which always tends to synchronize the oscillations of wave and particle. The escort wave model [2] replenishes the phase wave model by attributing a stochastic velocity component and a potential dependent rest mass to the particle. Moreover it introduces a novel method to deal with standing waves. In this new model a particle's trajectory is continuous but not deterministic because the particle is not only subjected to regular forces derived from external potentials but also to fluctuating forces. On this condition the space dependence of the wave escorting a particle agrees with the Schrödinger wave function. By a measurement process the phase harmony coupling is broken off. Contrary to the standard interpretation of quantum mechanics the escort wave does not collapse but disappears in the ocean of fluctuating waves, because it is no longer stabilized by the particle. The particle immediately starts to build up a new escort wave by adjusting its phase to the phases of fluctuating waves, whose frequencies and rest systems nearly agree with the corresponding magnitudes of the particle.