Approach to ergodicity in quantum wave functions

According to theorems of Shnirelman and followers, in the semiclassical limit the quantum wave functions of classically ergodic systems tend to the microcanonical density on the energy shell. Here we develop a semiclassical theory that relates the rate of approach to the decay of certain classical fluctuations. For uniformly hyperbolic systems, we find that the variance of the quantum matrix elements is proportional to the variance of the integral of the associated classical operator over trajectory segments of length T-H and inversely proportional to T-H(2), where T-H = h rho is the Heisenberg time, rho being the mean density of states. Since for these systems the classical variance increases linearly with T-H, the variance of the matrix elements decays like 1/T-H. For nonhyperbolic systems, such as Hamiltonians with a mixed phase space and the stadium billiard, our results predict a slower decay due to sticking in marginally unstable regions. Numerical computations supporting these conclusions are presented for the bakers map and the hydrogen atom in a magnetic field.