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arXiv:0704.1726

On de Broglie's quantum particle as the soliton solution of linear Schrödinger equation

Agus Budiyono

Published Apr 13, 2007• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.0704.1726

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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.
We consider a mass-less manifestly covariant {\it linear} Schrödinger equation. First, we show that it possesses a class of non-dispersive soliton solution with finite-size spatio-temporal support inside which the quantum amplitude satisfies the Klein-Gordon equation with finite {\it emergent} mass. We then proceed to interpret the soliton wave function as describing a particle with finite mass, energy and momentum. Inside the spatio-temporal support, the wave function shows spatio-temporal internal vibration with angular frequency and wave number that are determined by the energy-momentum of the particle as firstly conjectured by de Broglie. Imposing resonance of the internal vibration inside the spatio-temporal support leads to Planck-Einstein quantization of energy-momentum. The first resonance mode is shown to recover the classical energy-momentum relation developed in special relativity. We further show that the linearity of the Schrödinger equation allows one to construct many solitons solution through superposition, each describing a particle with various masses, energies and momenta.
Abstract-- We used a nonlinear Schrodinger equation to replace the linear Schrodinger equation and to study further the natures and states of microscopic particles due to plenty of difficulties of quantum mechanics. From this investigation we find that the states and properties of microscopic particles are considerably changed relative to those in quantum mechanics. An outstanding and obvious change is that the microscopic particles have a wave-corpuscle duality in such a case. The solution of the nonlinear Schrodinger equation contains an envelop and carrier waves with determinant frequency which can propagation in medium in a certain velocity. These display the wave feature of particle. However the solutions have all a mass centre and possess a determinant size and mass, momentum and energy, which satisfy also the conservation laws of mass, momentum and energy, at the same time, they meet the collision law of classical particles. These embody the corpuscle feature of the microscopic particles. Finally we seek the reasons and roots generating these changes, which are due to the nonlinear interactions among the particles or between the particles and background field to be considered in the nonlinear Schrodinger equation. The nonlinear interactions provide a double-well potential to make the microscopic particle to be localized as a soliton, and break also through the fundamental hypothesis for the independence of Hamiltonian
It has been shown that velocity of propagation of wave front cannot coincide with observable velocity of quantum particles. It is additional argument leads to conclusion that phase wave of de Broglie cannot be associated with single "elementary" particle like electron. Therefore quantum interference under linear superposition cannot describe energy distribution in extended quantum particles. Essentially new approach is required in order to establish non-linear relativistic wave equations with soliton-like solutions.
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.
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.
We study certain non-linear generalisations of the Schr{ö}dinger equation which admit static solitonic 2 solutions in absence of external potential acting on the particle. We consider a class of solutions that can be written as a product of a solution of the linear Schr{ö}dinger equation with a peaked quantum soliton, in a regime where the size of the soliton is quite smaller than the typical scale of variation of the linear wave. In the non-relativistic limit, the solitons obey a generalized de Broglie-Bohm (dB-B) guidance equation. In first approximation, this guidance equation reduces to the dB-B guidance equation according to which they move at the so-called de Broglie-Bohm velocity along the hydrodynamical flow lines of the linear Schr{ö}dinger wave. If we consider a spinorial electronic wave function a la Dirac, its barycentre is predicted to move exactly in accordance with the dB-B guidance equation.