Quantum Mechanical Time-Delay Matrix in Chaotic Scattering

We calculate the probability distribution of the matrix $Q=-iħS^{-1\mathrm{}}\partial S/\partial E$ for a chaotic system with scattering matrix $S$ at energy $E$. The eigenvalues ${\tau }_j$ 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.