Delay-time distribution in the scattering of time-narrow wave packets. (I)

TL;DR

This paper derives a comprehensive probability distribution for delay-times in wave packet scattering, connecting quantum, classical, and semi-classical regimes through a unified expression.

Contribution

It introduces a general formula for delay-time distribution in wave scattering that interpolates between quantum and classical limits, applicable to various wave systems.

Findings

In the monochromatic limit, delay-time matches Eisenbud-Wigner and Smith's expressions.

In the semi-classical limit, distribution aligns with classical mechanics.

The formula provides a smooth transition between quantum and classical delay-time regimes.

Abstract

This is the first of two subsequent publications where the probability distribution of delay-times in scattering of wave packets is discussed. The probability distribution is expressed in terms of the on-shell scattering matrix, the dispersion relation of the scattered beam and the wave packet envelope. In the monochromatic limit (poor time resolution) the mean delay-time coincides with the expression derived by Eisenbud and Wigner and generalized by Smith more than half a century ago. In the opposite limit, and within the semi-classical approximation, the resulting distribution coincides with the result obtained using classical mechanics or geometrical optics. The general expression interpolates smoothly between the two extremes. An application for the scattering of electromagnetic waves in networks of RF transmission lines will be discussed in the next paper to illustrate the method in an experimentally relevant context.