Analytic Continuation of Classical Mechanics for Classically Forbidden Collision Processes

Classically forbidden processes are those that cannot take place via ordinary classical dynamics. Within the framework of classical S-matrix theory, however, classical mechanics can be analytically continued and classical-limit approximations obtained for these classically forbidden, or weak transition amplitudes (i.e., S-matrix elements). The most powerful and general way of analytically continuing classical mechanics for a complex dynamical system is to integrate the equations of motion themselves through the classically inaccessible regions of phase space. Success in calculating these analytically continued trajectories is reported in this work; with certain special features of these complex-valued trajectories recognized and taken account of, it is seen that they are essentially as easy to deal with numerically as ordinary (i.e., real) classical trajectories. Application to the linear A+BC collision (vibrational excitation) gives excellent results; transition probabilities as small as 10−11 (the smallest ones available for comparison) have been obtained, agreement with the exact quantum mechanical values being within a few percent.