Quantum Mechanical Time-Delay Matrix in Chaotic Scattering

Abstract

We calculate the probability distribution of the matrix $Q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}i\ensuremath{\Elzxh}{S}^{\ensuremath{-}1}\ensuremath{\partial}S/\ensuremath{\partial}E$ for a chaotic system with scattering matrix $S$ at energy $E$. The eigenvalues ${\ensuremath{\tau}}_{j}$ of $Q$ are the so-called proper delay times, introduced by Wigner and Smith to describe the time dependence of a scattering process. The distribution of the inverse delay times turns out to be given by the Laguerre ensemble from random-matrix theory.