Group velocity of wave packets : a general formula

Ajoy Kumar Ghatak, S. Lokanathan

Wojciech T. Chyla

This paper presents the fully relativistic, general expression for the group velocity Vg of the electromagnetic wave packet in a relativistic dispersive medium. Motion of the medium is accounted for with the relativistic index of refraction. The general formula for Vg is specialized to a few particular situations that are briefly discussed. The explicit, exact, fully relativistic expression for Vg is relevant to all issues where the electromagnetic wave packet travels through a medium in motion, especially where the velocity of the medium is high or utmost accuracy is essential, e.g. it pertains to the slow light phenomenon, pulsar radiation, GPS, radar, and LIDAR technologies.

H. S. ALLEN

Qiong-Gui Lin

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.

Henry M. Bradford

The usual textbook discussions of the velocity and spreading of a wave packet are based on special cases and approximations which have a limited range of validity. In this paper more general relations are obtained by considering the time dependence of the moments of a wave packet. Both a packet of real sinusoidal plane waves (a real pulse, e.g., sound) and a packet of complex exponential plane waves (e.g., the quantum-mechanical wave function of a free particle) are considered. The results are similar if the real wave packet contains no standing wave components. It is shown that the centroid of a wave packet moves at a constant velocity which is a weighted average of the group velocity. The spreading of the wave packet depends on the product of the variance of the group velocity and the square of the time interval before or after the time of minimum packet width. The packet moves and spreads like a collection of particles in free flight, in which the velocities of the particles are the group velocities of the wave components of the packet.

K. T. Hecht