Propagation of General Wave Packets in Some Classical and Quantum Systems

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 thecenter-of-probability velocity of quantum mechanics, in spite of the fact that there exist in the literature at least eight different velocities forthe electromagnetic wave. We propose a center-of-energy velocity todescribe the entire motion of general wave packets in classicalphysical systems. It is a measurable quantity, and is well definedfor both continuous and discrete systems. For electromagnetic wavepackets it is a generalization of the velocity of energy transport.General wave packets in several classical systems are studied and thecenter-of-energy velocity is calculated and expressed in terms of thedispersion relation and the Fourier coefficients. These systemsinclude string subject to an external force, monatomic chain anddiatomic chain in one dimension, and classical Heisenberg model inone dimension. In most cases the center-of-energy velocity reduces tothe group velocity for quasi-monochromatic wave packets. Thus it alsoappears to be the generalization of the group velocity. Wave packetsof the relativistic Dirac equation are discussed briefly.