Nonperturbative Approach to the Theory of Transition Probabilities

A nonperturbative approach to the theory of transition probabilities is presented. The results of this theory are employed to investigate the validity of the assumptions inherent in the perturbative approach. For this purpose, the simple examples of absorption of radiation (first-order process) and Raman scattering (second-order process) are used. The method proposed, which is based upon finite-difference techniques, represents an accurate solution of the time-dependent Schr\"odinger equation in a given representation and has many advantages over methods based upon the perturbation series.