Energy-level statistics and time relaxation in quantum systems

We study a quantum-mechanical system, prepared, at t=0, in a model state, that subsequently decays into a sea of other states whose energy levels form a discrete spectrum with given statistical properties. An important quantity is the nln il al probability P(t), defined as the probability, at time t, to find the system in the original model state. Our main purpose is to analyze the influence of the discreteness and statistical properties of the spectrum on the behavior of P(t). Since P(t) itself is a statistical quantity, we restrict our attention to its ensemble average [P(t)], which is calculated analytically using random-matrix techniques. within certain approximations discussed in the text. We find, for [P(t)], an exponential decay, followed by a revival, governed by the two-point structure function of the statistical spectrum, thus giving a nonzero asymptotic value for large t's. The analytic result compares well with a number of computer simulations, over a time range discussed in the text.