Transition Probability due to Random Perturbations

The subject of transition probabilities induced by random perturbations is treated. The expression for average transition rates between two energy levels is derived for the general case when the frequency spectrum of the perturbation and the distribution of the energy levels have comparable widths. The general expression takes an especially simple form for the special cases where the frequency bandwidth of the perturbation Hamiltonian is either much larger or much smaller than that of the energy level distribution. The formulas are applied, by way of illustration, to a number of physical problems.