On the Theory of Wave Packets

TL;DR

This paper analyzes the dispersion of wave packets, comparing non-covariant and covariant Gaussian models, and provides a general formula for their time evolution, highlighting the physical interpretation of wave function disappearance.

Contribution

It introduces a covariant Gaussian wave packet model and derives a general dispersion formula, improving the understanding of wave packet evolution in relativistic contexts.

Findings

Non-covariant models predict too slow dispersion for relativistic particles.

Covariant models provide a more accurate description of wave packet dispersion.

Integral over time of flux and probability densities scales as 1/L^2, similar to classical particles.

Abstract

In this paper we discuss some aspects of the theory of wave packets. We consider a popular non-covariant Gaussian model used in various applications and show that it predicts too slow a longitudinal dispersion rate for relativistic particles. We revise this approach by considering a covariant model of Gaussian wave packets, and examine our results by inspecting a wave packet of arbitrary form. A general formula for the time dependence of the dispersion of a wave packet of arbitrary form is found. Finally, we give a transparent interpretation of the disappearance of the wave function over time due to the dispersion --- a feature often considered undesirable, but which is unavoidable for wave packets. We find, starting from simple examples, proceeding with their generalizations and finally by considering the continuity equation, that the integral over time of both the flux and probability densities are asymptotically proportional to the factor 1/L^2 in the rest frame of the wave packet, just as in the case of an ensemble of classical particles.