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arXiv:quant-ph/0605116

An interpretation of quantum mechanics based on a waveguide analogy

Roald Ekholdt

Published May 13, 2006• 0 citations
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Abstract

Based on an observation that the basic mode of a common microwave waveguide is a solution to the Klein-Gordon equation, quantum mechanics is modeled as the wave-function propagated inside a waveguide. The guide width is determined by the rest-energy and the potential energy. Moreover, a geometrical optics model of the propagation illustrates how the width as a function of distance determines the velocity and accelerations of the particle. The model entails the disappearance of action-at-a-distance problem, and that interactions are conveyed via the potential, i.e. by photons.

DOI
10.48550/arxiv.quant-ph/0605116

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Based on an observation that the basic mode of a common microwave waveguide is a solution to the Klein-Gordon equation, quantum mechanics is modeled as the wave-function propagated inside a waveguide. The guide width is determined by the rest-energy and the potential energy. Moreover, a geometrical optics model of the propagation illustrates how the width as a function of distance determines the velocity and accelerations of the particle. The model entails the disappearance of action-at-a-distance problem, and that interactions are conveyed via the potential, i.e. by photons.
Following the spirit of de Broglie and Einstein, we think the concepts of matter and radiation can be unified. We know a particle propagates like a wave; its motion is described by certain wave equations. At this point, it is not clear what the wave function represents. Besides the statistical meaning suggested by the Copenhagen interpretation, does the wave function represent any physical motion? For photon, we know it is an electro-magnetic wave. But what about particles with rest mass, such as an electron? To investigate the physical nature of matter wave, we propose that: (1) Like the photon, a particle is an excitation wave of a real physical field. (2) Different types of particles are different excitation modes of the same field. Based on this thinking, we show that the concept of quantum mechanics can be a natural extension of classical mechanics. By critically analyzing the transition from classical physics to quantum physics, we found a new physical meaning for the quantum wave function. This work suggests that various quantum wave equations, including the Klein-Gordon equation, the Dirac equation and the Schrodinger equation, could have a common base relating to the Basic Wave Equation of the matter wave. This work has some interesting implications. It suggests a possible way to explain the origin of visible matters and dark matters in our universe.
A derivation of the Klein-Gordon wave equation without substituting the corresponding quantum mechanical energy and momentum operators in Einstein's relativistic energy-momentum relationship is presented. An additional discussion is made to motivate readers to understand the distinction between the quantum wave equation and the governing law in the context of quantum mechanical formalisms. A resemblance of mathematical structure between the Maxwell wave equation and the deduced Klein-Gordan equation is shown. It is also noted that the wave velocity of the matter wave becomes equal to photon velocity for massless cases. This can be useful to explain why light equally shows matter and particle behaviour.
Expressions are derived for the electromagnetic fields of guided waves which are analogous to the quantum-mechanical equations representing barrier tunneling. This analogy is achieved by comparing the propagation constant of the Schrödinger time-independent wave equation with that of the electromagnetic wave equation in waveguide and by comparing the de Broglie wavelength of a particle with the wavelength of the waves that propagate the energy. This results in an expression relating the form of an arbitrary one-dimensional energy barrier to the physical dimensions of a section of waveguide. The analogy is tested by the propagation of energy in the TE10 mode at both the 3- and 6-cm bands for the cases of rectangular and hyperbolic barriers. Although evanescent modes are present at the discontinuous regions, the analog for the rectangular barrier, which is considered to be the worse case, is verified when an effective barrier length leff of about 1.2l is used. This experimental verification demonstrates the possibility of waveguide simulation of quantum-mechanical energy barriers and the practicality of utilizing an electromagnetic analog for demonstrating the tunneling phenomenon and provides a method for measurement of the transmission coefficient through an arbitrarily shaped barrier.
The motion of a particle in a potential channel (two-dimensional well without a potential gradient along the third coordinate) is characterized by a discrete set of allowed modes or travelling Ψ waves. A paraxial potential channel can transmit the initial distribution of the Ψ function to channel cross sections some distance from the entrance. The dispersion equation for a channel having transverse dimensions of the order of the Compton wavelength is the same as the free motion equation, the eigenvalue of the lower channel mode being identified with the rest energy of the particle.