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.

## REFERENCES

*Quantum Mechanics*(McGraw‐Hill, New York, 1968), pp. 278–279.

*Classical Mechanics*(Addison‐Wesley, Reading, Mass., 1950), pp. 288–294.

*Quantum Mechanics and Path Integrals*(McGraw‐Hill, New York, 1965).

*Methods of Theoretical Physics*(McGraw‐Hill, New York, 1953), pp. 434–443.

*Quantum Mechanics*(Wiley, New York, 1961), pp. 222–224;

*The Fundamental Principles of Quantum Mechanics*(Dover, New York, 1958), pp. 24–26, 43–46.

*S*,

*Scattering Theory of Waves and Particles*(McGraw‐Hill, New York, 1966), pp. 167–175. Furthermore, since for the pair of action‐angle variables (

*n*,

*q*) it is

*n*that is the observable and must therefore be real,

*q*is completely unspecified and may therefore be complex.

*Introduction to Numerical Analysis*(McGraw‐Hill, New York, 1956), pp. 373–375.

*The Padé Approximant in Theoretical Physics*, edited by G. A. Baker and J. L. Gammel (Academic, New York, 1970).

*maximum*, is then easily estimated. A second batch of real trajectories is similarly computed, these with initial conditions $n1\u2009=\u20091$ and $q1$ swept through the (0, 2π) interval; from these trajectories $qmin,$ the value for which the final quantum number is a

*minimum*is estimated. The real parts of $q1$ and $q2,$ the roots of Eq. (B4), are then approximately $qmax$ and $(2\pi \u2212qmin),$ respectively, and

*any*guess for the imaginary parts suffices to start the iterative root‐finding procedure. (The real part of $q2$ is $2\pi \u2212qmin$ because the trajectory with

*initial*conditions $n1\u2009=\u20091$ and $q1\u2009=\u2009qmin$ is physically identical to the trajectory with

*final*conditions $n2\u2009=\u20091$ and $q2\u2009=\u20092\pi \u2212qmin$).