Exponential decay, recurrences, and quantum-mechanical spreading in a quasicontinuum model

We consider a single quantum state coupled equally to each of a set of evenly spaced quasicontinuum (QC) states. We obtain a delay differential equation for the initial-state probability amplitude, and this equation is solved analytically. When the QC-level spacing goes to zero, the initial-state probability decays exactly exponentially. For finite QC-level spacings, however, there are recurrences of initial-state probability. We discuss Tolman's "quantum-mechanical spreading" of probability and also a classical analog of our model.