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arXiv:physics/0404044

What is rest mass in the wave-particle duality? A proposed model

Donald C. Chang

Published Apr 8, 2004• 2 citations
arXivHTMLPDF

Abstract

In the wave-particle duality, a free particle can be considered as a wave packet. Is there a wave property that corresponds to the rest mass of the particle? This is an interesting question that has not been extensively explored before. We suggest that this problem may be approached by treating the rest mass on the same footing as energy and momentum. Here we demonstrate that, by assuming that the matter wave of a particle behaves like a photon, one could derive the mass-energy relation from the solution of a simplified wave equation. This solution suggests that the rest mass of a particle is associated with a "transverse wave number", which characterizes the radial variation of the wave function in the transverse plane. This finding has several appealing features. For example, it shows a consistent geometrical relationship between mass, energy and momentum in both the wave and particle perspectives. Also, it predicts that a massless particle must travel at the constant speed of light. Furthermore, from a philosophical point of view, this model suggests that the mass-energy relation attributed to the special theory of relativity may have a root in the wave-particle duality. Indeed, it is shown that the Klein-Gordon equation can be derived naturally based on this model.

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In the wave-particle duality, a free particle can be considered as a wave packet. Is there a wave property that corresponds to the rest mass of the particle? This is an interesting question that has not been extensively explored before. We suggest that this problem may be approached by treating the rest mass on the same footing as energy and momentum. Here we demonstrate that, by assuming that the matter wave of a particle behaves like a photon, one could derive the mass-energy relation from the solution of a simplified wave equation. This solution suggests that the rest mass of a particle is associated with a "transverse wave number", which characterizes the radial variation of the wave function in the transverse plane. This finding has several appealing features. For example, it shows a consistent geometrical relationship between mass, energy and momentum in both the wave and particle perspectives. Also, it predicts that a massless particle must travel at the constant speed of light. Furthermore, from a philosophical point of view, this model suggests that the mass-energy relation attributed to the special theory of relativity may have a root in the wave-particle duality. Indeed, it is shown that the Klein-Gordon equation can be derived naturally based on this model.
A quaternionic mass defining longitudinal mass () and transverse mass () of a particle is introduced. The longitudinal and transverse masses represent the mass of the particle and matter wave accompanying it, respectively. A massive particle is found to have a bigger longitudinal mass than transverse mass. The particle nature due to the transverse mass of a particle is described by a scalar wave satisfying the Klein-Gordon equation. The transverse mass vector is found to be along the direction of the particle’s velocity (). A particle of an equal longitudinal and transverse mass travels at speed of light in vacuum. The transmission of matter wave energy is found to be opposite to the particle velocity. The momentum of the particle () and its matter wave () are equal. The direction of the transverse mass is found to be along the direction of the particle’s velocity.
It is well known that the mass of a particle has properties different from Newtonian mechanics. First, it is speed-dependent. Second, it is convertible to energy. These properties were generally thought to be derived from the principle of relativity (PR). We have conducted a careful examination of the historical records and found that the non-Newtonian properties of mass were derived not so much based on PR, but more based on Einstein’s intuitive thinking that radiation and matters behave similarly. This gives us a hint: Since both photon and electron can behave as a particle as well as a wave, can such a wave nature account for the deviations from Newtonian mechanics? Thus, we have developed a wave model to describe the motion of a free particle with or without rest mass. We found that both the speed-dependence of mass and the mass-energy equivalence can indeed be derived based on the wave properties of a particle. This wave hypothesis has several advantages; it can naturally explain why particles can be created in the vacuum and why a particle cannot travel faster than the speed of light.
The wave-particle duality of quantum mechanics has always been an unsolved problem in physics. This article attempts to use one of the properties to explain the other, so as to eliminate the confusion of quantum mechanics probability waves. Specifically, this article finds that the discreteness of energy is the inherent property of all waves, so this article explains the particle nature of light from the perspective of light wave, thereby eliminating the confusion of light’s wave-particle duality; in addition, this article found that microscopic matter particles are only suitable for discussing the number of scattered particles or energy flow density, not their position and momentum, when they are forced to discuss their position and momentum, it will inevitably lead to confusion about probability waves, when only discussing the number of scattered particles or energy flow density, its volatility can be explained from the perspective of particle nature, thereby eliminating the confusion of microscopic matter particle probability waves. When the attribute of light as a wave is established, the light needs to overcome the Hamiltonian of the medium in different constraint systems (gravitational fields) during the propagation process, which will produce a universal redshift phenomenon, this can provide a new understanding of cosmic redshift; when the property of light as a wave is established, it means that the speed of light is a constant speed relative to the medium, which can provide a new understanding of the principle of constant speed of light.
A satisfactory explanation of the confounding wave-particle duality of matter is presented in terms of the reality of the wave nature of a particle. In this view, a quantum particle is an objectively real wave packet consisting of irregular disturbances of underlying quantum fields. It travels holistically as a unit and thereby acts as a particle. Only the totality of the entire wave packet at any instance embodies all the conserved quantities, for example the energy-momentum, rest mass, and charge of the particle, and as such must be acquired all at once during detection. On this basis, many of the bizarre behaviors observed in the quantum domain, such as wave function collapse, the limitation of prediction to only a probability rather than an actuality, the apparent simultaneous existence of a particle in more than one place, and the inherent uncertainty can be reasonably comprehended. The necessity of acquiring the wave function in its entirety for detection, as evinced by the appearance of collapse of the wave function, supports the paradigm of reality of the wave function described here. Quanta 2016; 5: 93–100.
In general, it is well known that the Schrödinger equation is not compatible with special relativity theory to a necessary extent.Thus far, there are already several trials to formulate versions of the Schrödinger equation to ensure compatibility with special relativity theory, the Klein-Gordon-equation or the Dirac equation are some of these attempts.Material and Methods: In this paper, Einstein's relativistic energy momentum relation is re-analyzed, a normalized relativistic energy momentum relation is derived.The derived normalized relativistic energy momentum relation together with the known Schrödinger equation is used as a starting point to establish a wave equation consistent with special relativity theory.Results: In this publication, based on Einstein's relativistic energy momentum relation, the historical problem of the "particle-wave-duality" is solved.Furthermore, a special relativity theory consistent wave equation is derived.
In this paper, we have considered that the particle velocity in the matter wave concept of de Broglie is relative. It reveals that matter wave associated with a moving particle is a photon wave which has longitudinal and transverse values. The conventional de-Broglie’s wavelength-momentum relation corresponds to the longitudinal wave. As a consequence, we have found the longitudinal and transverse relative values of the mass, time, length and velocity including the missing equations of Einstein’s theory of special relativity. It also reveals that momentum and energy are relative quantities and the corresponding expressions for the longitudinal and transverse values for both of them have been presented. Moreover, the addition rules, with a general expression, for all the relative quantities based on Lorentz-Einstein transformation law for the addition of velocities have been given. Finally, the maximum and minimum values of the relative quantities have been expressed in terms of Plank’s constant and the speed of light. Their numerical values have also been calculated and presented in this paper.