Coherent Wavepackets of Phonons

Abstract

The propagation of an elastic pulse in a one-dimensional elastic medium is investigated. First, normal modes and frequencies are introduced to characterize the elastic medium, consisting of a line of atoms harmonically vibrating around a stahle equilibrium configuration. Second, the equation of motion of an elastic disturbance along the chain is obtained in the classical limit (d’Alembert equation) and the solution corresponding to a single elastic pulse is commented. Third, the quantum mechanical treatment is formulated and the pulse propagation is written in terms of phonons: the pulse is seen to correspond to a coherent wavepacket of phonons propagating at the “sound” velocity of the chain. Different coherent wave-packets are considered and their change during the motion is related to their composition in energy and wavevectors. Finally, “birth and death” of the pulse are discussed.

Two appendices are added at the end to collect in a compact way some properties of the coherent states (Appendix A) and useful relations dealing with the Poisson distribution (Appendix B).