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arXiv:physics/0609216

A class of traveling-envelope solutions of free Schrödinger equation generated by Lorentz transformation

Agus Budiyono

Published Sep 24, 2006• 0 citations
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Abstract

We develop a class of traveling-envelope solutions of Schrödinger equation for a free particle whose amplitude is moving with constant group velocity while keeping its shape undistorted. We show that solution with arbitrary finite group velocity is obtained by Lorentz boosting the solution with vanishing group velocity, if the quantum average energy $E$ and momentum $p$ are related to the rest-mass $m$ of the particle by Einstein formula $E^2/c^2-p^2=m^2c^2$. The wave function is spatially localized with finite-size support which is decreasing as the rest-mass and/or group velocity are increased. For a particle with vanishing rest-mass yet finite momentum, we show that the group and phase velocities are equal to the velocity of light and the wavelength is given by Einstein another formula $λ_P=h/p$.

DOI
10.48550/arxiv.physics/0609216

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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.
When the solution of the time independent Schrödinger equation is a real function, the velocity of the associated quantum particle is vanishing in Bohmian Mechanics. We can obtain a moving particle if we consider that its velocity can be related not to the phase of the wave function but to its amplitude. By using the rules of standard quantum mechanics, we have constructed a model generalizing the momentum operator and modifying the corresponding Schrödinger equation. We obtain a new nonlinear equation of the form studied by Ermakov and Pinney and we apply the model to the cases of the particle in a box and of a bouncing ball. The new formula of quantum velocity and the corresponding motion of quantum particles have an immediate correspondence with the behavior of the bodies in classical mechanics.
It is well-known that Schrodinger wave functions are not covariant under Galilean boosts. To obtain the correct result the boost transformation, t′ = t and ~x′ = ~x − ~vt, must be followed by the phase shift δφ = 12mv 2t + m~v · ~r. A generally accepted approach is to absorb the phase shift into the Galilean boost, construct the Schrodinger group and claim Galilean invariance of the Schrodinger wave function. Here I address the physical meaning of the phase shift. It is not a coordinate transformation since it depends on the mass of the Schrodinger particle. Consequently, one needs as many Schrodinger groups as there are distinct masses. The phase shift does not follow from Lorentz boost per se in the low velocity limit. Covariance of the non-relativistic quantum mechanical kinetic energy and momentum under pure coordinate transformations can be satisfied only by the boost t′ = t ( 1 + 12 v2 c2 + ~v·~r c2 )
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.
In this article, it is shown that if a particle is at rest and has some harmonic oscillation, then a traveling particle with a constant speed corresponds to a De Broglie Wave.In 1924, De Broglie proposed that particles of matter exhibit wave like behavior, the expression for which became fundamental in the creation of wave mechanics (quantum mechanics).The main equation for this is the Schrödinger equation, which in the case of free motion of a particle becomes the expression for the De Broglie wave.This hypothesis is supported by many early experiments on the diffraction of electrons, and later the development of modern physics.However until now, the origin of this hypothesis remains unclear.We show that the expression for the De Broglie wavelength can be obtained from a more general well-known concept, that any particle at rest has a corresponding harmonic oscillation that transforms into a wave of a moving particle with the help of relativistic change of coordinates (the Lorentz transformations).In this way, a De Broglie wave can be explained as a spacetime wave.
The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schrödinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schrödinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schrödinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schrödinger equation.