Variance of Quantum Matrix Elements

Abstract

The fluctuations of quantum matrix elements has been examined recently of Eckhardt et al. (1995) and Eckhardt and Main (1995). For hyperbolic systems in the semiclassical approximation, they show that the variance decays as the inverse of the Heisenberg time and that the classical and quantum fluctuations are related, as will be described below. In a little more detail, the conjectures state that the fluctuations of the diagonal matrix elements around the mean value, < F ² _j >=< (A _jj -Ā)² >, of a quantum mechanical observable have the same order of magnitude as the mean square of the off diagonal terms < |A _jk |² > and they decrease proportional to the inverse of Heisenberg time as the semiclassical limit is taken.