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.

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W. H.
Miller
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C. C.
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A short review of the early part of this work is in
W. H.
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R. A.
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[
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4.
See, for example, L. I. Schiff, Quantum Mechanics (McGraw‐Hill, New York, 1968), pp. 278–279.
5.
W. H. Miller and T. F. George, “Semiclassical Theory of Electronic Transitions in Low Energy Atomic and Molecular Collisions Involving Several Nuclear Degrees of Freedom,” J. Chem. Phys. (to be published).
6.
H. Goldstein, Classical Mechanics (Addison‐Wesley, Reading, Mass., 1950), pp. 288–294.
7.
Although we think of the internal (i.e., quantized) degrees of freedom semiclassically in terms of their action‐angle variables, the numerical integration of the equations of motion is actually carried out in ordinary Cartesian coordinates. For the linear A+BC system, of Ref. 23, for example, if n1 and q1 are the initial action‐angle variables, one converts these into initial conditions for the Cartesian variables
r = (2n1+1)1/2cosq1,
p1 = −(2n1+1)1/2sinq1,
and the trajectory is integrated with the classical Hamiltonian H(P,R,p,r) = P2/2m+p2/2+r2/2+exp[α(r−R)]. At the conclusion of the trajectory the final values r2 and p2 are used to construct the final values of the action‐angle variables
n2 = 12(p22+r22)−12,
q2 = −tan−1(p2/r2).
8.
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw‐Hill, New York, 1965).
9.
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw‐Hill, New York, 1953), pp. 434–443.
10.
C.
Morette
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I. M.
Gel’fand
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A. M.
Yaglom
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12.
H. Goldstein, Ref. 6, pp. 36–38.
13.
See, for example, (a) A. Messiah, Quantum Mechanics (Wiley, New York, 1961), pp. 222–224;
and (b) E. C. Kemble, The Fundamental Principles of Quantum Mechanics (Dover, New York, 1958), pp. 24–26, 43–46.
14.
H. Goldstein, Ref. 6, pp. 273–284.
15.
It is easy to see that the “formal theory of scattering” requires only that Re(t2−t1)→+∞,Im(t2−t1) finite, in defining the operator S,
S = exp(iHot2/ℏ) ezp[−iH(t2−t1)/ℏ] exp(−iHot1/ℏ);
see, for example, R. G. Newton, 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.
16.
F. B. Hildebrand, Introduction to Numerical Analysis (McGraw‐Hill, New York, 1956), pp. 373–375.
17.
The phase φ[n2(q1,n1),n1]≡φ(q1) is actually not periodic: φ(q1+2π) = φ(q1)−2π[n2(q1,n1)−n1], but χ(q1)≡φ(q1)+q1[n2(q1,n1)−n1] is; therefore, it is the function χ(q1) that is actually numerically analytically continued, and then φ is calculated by φ(q1) = χ(q1)−(n2−n1)q1.
18.
The Padé Approximant in Theoretical Physics, edited by G. A. Baker and J. L. Gammel (Academic, New York, 1970).
19.
Periodic functions such as n2(q1) can also be represented by the point method using continued fractions; see, for example, Ref. 20. The basis functions must be chosen, however, so that their zeros occur only at the ends of the region of periodicity. Thus we found that n2(q1) could be represented by a linear combination of the basis functions uk(x) = sink(x/2). The differences x−xi in the continued fraction expressions [see Eqs. (2.9) and (2.10) of Schlessinger, Ref. 20] are replaced by sin[(x−xi)/2], where xi is an input point. The procedure described in the text, however, being a more natural extension of the well‐known Fourier Series, seemed preferable to us.
20.
L.
Schlessinger
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21.
All of the classically forbidden transitions of Ref. 1(c) were also recalculated using the procedure of this section, and the earlier results were confirmed to within at least one unit in the last significant figure reported there.
22.
For derivation and discussion of the uniform semiclassical formulas, see Refs. 1(b) and 2(c).
23.
D.
Secrest
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24.
The equation Imn2(q1,n1) = 0 defines a line in the complex q1 plane. There is thus zero probability that successive q1 iterates give final values n2(q1) that are real.
25.
See, for example, F. B. Hildebrand Ref. 16, pp. 233–239.
26.
F. B. Hildebrand, Ref. 16, pp. 38–45.
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W. H.
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K.
Freed
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32.
A straightforward secant version of the Newton‐Raphson iteration scheme (i.e., multidimensional linear interpolation) converged to the root of Eq. (B4) with typically only 5 to 6 iterations, one complex trajectory being required for each alteration. To achieve the three‐figure accuracy reported in Table II it was necessary to specifiy only normal error limits (e.g., ε≃10−6) for this root‐finding operation, as well as the trajectory integration itself, i.e., repetition of the over‐all procedure with lower error limits (e.g., ε≃10−8) caused no changes to at least three significant figures. The root‐finding procedure was fairly insensitive to initial guesses for the roots of Eq. (B4) but did, of course, require fewer iterations if reasonable initial guesses were used. Good initial guesses for the roots could be obtained quite easily by the following procedure: For a 0→1 transition, for example, one computes a small batch (e.g., five) of ordinary real trajectories with initial conditions n1 = 0 and q1 swept through its (0, 2π) interval at five equally spaced points; qmax, the value of q1 for which the final quantum number is a maximum, is then easily estimated. A second batch of real trajectories is similarly computed, these with initial conditions n1 = 1 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 (2π−qmin), respectively, and any guess for the imaginary parts suffices to start the iterative root‐finding procedure. (The real part of q2 is 2π−qmin because the trajectory with initial conditions n1 = 1 and q1 = qmin is physically identical to the trajectory with final conditions n2 = 1 and q2 = 2π−qmin).
33.
F. B. Hildebrand, Ref. 16, pp. 188–202.
34.
R. Van Wyk, “Variable Mesh Methods for Differential Equations,” NASA Report CR‐1247, November 1968.
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