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arXiv:0803.0467

A spatial envelope soliton model of the electron - propagating within a fictitious waveguide

Roald Ekholdt

Published Mar 4, 2008• 0 citations
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Abstract

The paper proposes an envelope soliton model of the electron. The soliton propagates within a waveguide which size varies with the potential. The soliton is the solution of the Schroedinger equation with the addition of a nonlinear term that is the negative of Bohm's Quantum potential. This potential is interpreted as representing the dispersion of the linear equation. The model is based on de Broglie's original wave-particle concept, which is based on the Special theory of relativity. The model thus illustrates that theory.

DOI
10.48550/arxiv.0803.0467

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The paper proposes an envelope soliton model of the electron. The soliton propagates within a waveguide which size varies with the potential. The soliton is the solution of the Schroedinger equation with the addition of a nonlinear term that is the negative of Bohm's Quantum potential. This potential is interpreted as representing the dispersion of the linear equation. The model is based on de Broglie's original wave-particle concept, which is based on the Special theory of relativity. The model thus illustrates that theory.
The paper proposes an envelope soliton model of the electron that propagates as a protuberance on a fictitious waveguide, which acts as trajectory. The model is based on de Broglie's original electron wave-particle relativistic theory, and on the observation that the Klein-Gordon equation governs the propagation mode of a common microwave waveguide - with the Schroedinger equation as a low group velocity approximation. This analogy opened for practical physical models, including solitons. In the linear case a conceived corpuscle zigzags within the fictitious waveguide - a zigzagging that resembles Penrose's picture of Dirac's electron theory. The soliton envelope is defined by a special nonlinear version of the Schroedinger equation. The nonlinearity cancels the de Broglie/Bohm Quantum Potential of the envelope. The wavefunction is confined by the envelope, and thus the concept of the collapse of the wavefunction is eliminated. Since the electron model is based directly on the Special theory of relativity it also illustrates that theory. The corpuscle, incorporating spin and charge, which moves at the speed of light, needs further study relating it to the photon. A model of the photon based on the mutual coupling of two modes of an optical fiber at the Planck scale, is suggested. One of these modes is thought to carry two orthogonal polarized electromagnetic external fields, while the other, internal, mode carries a corpuscle that moves helically and thus represents the spin. A single electron is considered to have one polarization only, without any superposition possibility; hence, the concept of collapse in a polarization measurement is eliminated.
This article proposes an interpretation of quantum physics based on the theory of solitons. According to this interpretation, the elementary particle (in particular, the electron) is a soliton solution of the system of nonlinear equations, while the linear equations of quantum mechanics for the wave functions represent the boundary conditions for soliton solutions. The nonlinear equations for the quantum electron are hypothesized to be the usual Maxwell equations in which the charge and current densities are expressed through quadratic combinations of the electromagnetic field strength. The complex wave function describing the motion of the electron in this case is the usual electromagnetic wave, where the real part is the electric field strength, and the imaginary part is the magnetic field strength. Soliton equations, Maxwell equations and quantum equations are easily written using 3+1 Pauli matrices, which indicates that the 3+1 system of coordinates of space and time is a natural realization of the particle-wave soliton world. The proposed interpretation allows combining both the Copenhagen interpretation and Bohm's theory of "hidden" variables.
The photon is modeled as a monochromatic solution of Maxwell’s equations confined as a soliton wave by the principle of causality of special relativity. The soliton travels rectilinearly at the speed of light. The solution can represent any of the known polarization (spin) states of the photon. For circularly polarized states the soliton’s envelope is a circular ellipsoid whose length is the observed wavelength (λ), and whose diameter is λ/π; this envelope contains the electromagnetic energy of the wave (hv = hc/λ). The predicted size and shape is confirmed by experimental measurements: of the sub‐picosecond time delay of the photo‐electric effect, of the attenuation of undiffracted transmission through slits narrower than the soliton’s diameter of λ/π, and by the threshold intensity required for the onset of multiphoton absorption in focussed laser beams. Inside the envelope the wave’s amplitude increases linearly with the radial distance from the axis of propagation, being zero on the axis. Outside the envelope the wave is evanescent with an amplitude that decreases inversely with the radial distance from the axis. The evanescent wave is responsible for the observed double‐slit interference phenomenon.
We develop a class of soliton solution of {\it linear} Schrödinger equation without external potential. The quantum probability density generates its own boundary inside which there is internal vibration whose wave number is determined by the velocity of the particle as firstly conjectured by de Broglie. Assuming resonance of the internal vibration will lead to quantization of particle's momentum in term of wave number of the envelope quantum probability density. We further show that the linearity of the Schrödinger equation allows us to have non-interacting many solitons solution through superposition, each describing a particle with equal mass.
The photon is modeled as a monochromatic solution of Maxwell's equations confined as a soliton wave by the principle of causality of special relativity. The soliton travels rectilinearly at the speed of light. The solution can represent any of the known polarization (spin) states of the photon. For circularly polarized states the soliton's envelope is a circular ellipsoid whose length is the observed wavelength (A), and whose diameter is λ/π; this envelope contains the electromagnetic energy of the wave (hv = hc/λ). The predicted size and shape is confirmed by experimental measurements: of the sub-picosecond time delay of the photo-electric effect, of the attenuation of undiffracted transmission through slits narrower than the soliton's diameter of λ/π, and by the threshold intensity required for the onset of multiphoton absorption in focussed laser beams. Inside the envelope the wave's amplitude increases linearly with the radial distance from the axis of propagation, being zero on the axis. Outside the envelope the wave is evanescent with an amplitude that decreases inversely with the radial distance from the axis. The evanescent wave is responsible for the observed double-slit interference phenomenon.
The photon is modeled as a monochromatic solution of Maxwell's equations confined as a soliton wave by the principle of causality of special relativity. The soliton travels rectilinearly at the speed of light. The solution can represent any of the known polarization (spin) states of the photon. For circularly polarized states the soliton's envelope is a circular ellipsoid whose length is the observed wavelength (λ), and whose diameter is λ/π; this envelope contains the electromagnetic energy of the wave (hν = hc/λ). The predicted size and shape is confirmed by experimental measurements: of the sub-picosecond time delay of the photo-electric effect, of the attenuation of undiffracted transmission through slits narrower than the soliton's diameter of λ/π, and by the threshold intensity required for the onset of multiphoton absorption in focussed laser beams. Inside the envelope the wave's amplitude increases linearly with the radial distance from the axis of propagation, being zero on the axis. Outside the envelope the wave is evanescent with an amplitude that decreases inversely with the radial distance from the axis. The evanescent wave is responsible for the observed double-slit interference phenomenon.