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arXiv:physics/9809033 · Annalen der Physik

Quantile motion of electromagnetic waves in wave guides of varying cross section and dispersive media

H.D. Dahmen, E. Gjonaj, T. Stroh

Published Dec 1, 1998• 4 citations
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Abstract

We discuss applications of the quantile concept of trajectories and velocities to the propagation of electromagnetic signals in wave guides of varying cross section. Quantile motion is a general description of the transport properties of measurable quantities in quantum mechanics as well as in classical electrodynamics. In the latter case we consider the quantile motion of electromagnetic energy as the direct result of a physical measurement. In this sense the quantile velocity corresponds to the electromagnetic signal velocity. We show that this signal velocity is always smaller than the speed of light in vacuum.

DOI
10.1002/(sici)1521-3889(199812)7:7/8<645::aid-andp645>3.0.co;2-p

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We discuss applications of the quantile concept of trajectories and velocities to the propagation of electromagnetic signals in wave guides of varying cross section. Quantile motion is a general description of the transport properties of measurable quantities in quantum mechanics as well as in classical electrodynamics. In the latter case we consider the quantile motion of electromagnetic energy as the direct result of a physical measurement. In this sense the quantile velocity corresponds to the electromagnetic signal velocity. We show that this signal velocity is always smaller than the speed of light in vacuum.
Abstract We discuss applications of the quantile concept of trajectories and velocities to the propagation of electromagnetic signals in wave guides of varying cross section. Quantile motion is a general description of the transport properties of measurable quantities in quantum mechanics as well as in classical electrodynamics. In the latter case we consider the quantile motion of electromagnetic energy as the direct result of a physical measurement. In this sense the quantile velocity corresponds to the electromagnetic signal velocity. We show that this signal velocity is always smaller than the speed of light in vacuum.
We discuss applications of the quantile concept of trajectories and velocities to the propagation of electromagnetic signals in wave guides of varying cross section. Quantile motion is a general description of the transport properties of measurable conserved quantities in quantum mechanics as well as in classical electrodynamics. In the latter case we consider the quantile motion of electromagnetic energy as the direct result of a physical measurement. In this sense the quantile velocity corresponds to the electromagnetic signal velocity also in the presence of barriers and inhomogeneities in the medium of propagation. We show that this signal velocity is always smaller than the speed of light in vacuum. Using numerical examples we demonstrate how typical wave phenomena can be described in terms of the quantile motion.
For electromagnetic propagation in linear dispersive media, two definitions of energy transport velocity may be introduced. The first is the transport velocity of total stored energy density, and it was first discussed by Brillouin. This concept does not distinguish different forms of energy, but considers only the total. It is equal to the group velocity for the case of a lossless dispersive medium. Its significance is not fully understood for a maser medium where it is either greater than the velocity of light or negative, or for a dissipative dispersive medium where it does not seem to offer a simple description of the propagation process. The second is the transport velocity of electromagnetic energy. It is based on the distinction of propagating energy which is just electric and magnetic field energy, from localized energy which is stored in the degrees of freedom of the dispersive medium. Thus propagation involves electromagnetic propagation per se, plus a transfer of energy between the propagating and localized forms. This transport velocity is always less than the velocity of light, for absorbing and maser media alike. It may be determined through a gain or attenuation measurement since it enters the gain equation of electromagnetic transmission structures.
In quantum mechanics the center of a wave packet is precisely defined as the center of probability. The center-of-probability velocity describes the entire motion of the wave packet. In classical physics there is no precise counterpart to the center-of-probability velocity of quantum mechanics, in spite of the fact that there exist in the literature at least eight different velocities for the electromagnetic wave. We propose a center-of-energy velocity to describe the entire motion of general wave packets in classical physical systems. It is a measurable quantity, and is well defined for both continuous and discrete systems. For electromagnetic wave packets it is a generalization of the velocity of energy transport. General wave packets in several classical systems are studied and the center-of-energy velocity is calculated and expressed in terms of the dispersion relation and the Fourier coefficients. These systems include string subject to an external force, monatomic chain and diatomic chain in one dimension, and classical Heisenberg model in one dimension. In most cases the center-of-energy velocity reduces to the group velocity for quasi-monochromatic wave packets. Thus it also appears to be the generalization of the group velocity. Wave packets of the relativistic Dirac equation are discussed briefly.
A novel definition of pulse propagation velocity is introduced.It is shown that the present definition does not lead to confusing results such as complex velocity or velocity exceeding the light velocity in the vacuum.Also shown are the parallels of this definition to the classical and quantum mechanics conceptions.Using the present definition reveals certain analogies between electromagnetic pulse propagation in the classical physics and de Broglie wave-packets propagation in the quantum mechanics, thus adding support to its validity.