Random Matrix Elements and Eigenfunctions in Chaotic Systems

The expected root-mean-square value of a matrix element Aαβ in a classically chaotic system, where A is a smooth, h̄-independent function of the coordinates and momenta, and α and β label different energy eigenstates, has been evaluated in the literature in two different ways: by treating the energy eigenfunctions as gaussian random variables and averaging |Aαβ| 2 over them; and by relating |Aαβ | 2 to the classical timecorrelation function of A. We show that these two methods give the same answer only if Berry’s formula for the spatial correlations in the energy eigenfunctions (which is based on a microcanonical density in phase space) is modified at large separations in a manner which we previously proposed. Typeset using REVTEX ∗ E–mail: [email protected] † E–mail: [email protected]