The probability of vibrational energy exchange in a molecular collision can be calculated using (1) a wave‐mechanical treatment using the method of ``distorted waves,'' (2) a semiclassical time‐dependent perturbation procedure in which the perturbation energy is obtained as a function of time from the classical collision trajectory, and (3) a purely classical calculation of the energy transferred to a classical vibrator. These methods are reviewed, related, and compared.
REFERENCES
1.
2.
3.
In Refs. 4 and 5, Herzfeld identifies L of the classical theory with 2πL of the wave‐mechanical theory because of this misunderstanding.
4.
K. F. Herzfeld, “Relaxation Phenomena in Gases,” in Thermodynamics and Physics of Matter, edited by F. Rossini (Princeton University Press, Princeton, New Jersey, 1955).
5.
K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic Press Inc., New York, 1959).
6.
7.
D. Rapp, M.S. thesis, Princeton University, 1956.
8.
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The proof given was also worked out independently by R. A. Allen and P. Feuer (private communication). It is interesting to speculate whether one could use the WKB wavefunctions directly in Eq. (23) to obtain a simple result which is intermediate in range of validity between the wave‐mechanical and semiclassical results. This author has been unable to effect the integration of the matrix element with these wavefunctions. However, Landau [L. Landau and E. Lifshitz, Quantum Mechanics (Pergamon Press, London, 1958), pp. 178–183] has arrived at a principle
[see also
B.
Widom
, Discussions Faraday Soc.
33
, 37
(1962
)] which allows one to evaluate the exponential part of the matrix element. Landau’s procedure is roughly equivalent to the following discussion given in this paper, although the details are somewhat nebulous to this author.13.
14.
J. C. Slater and N. H. Frank, Mechanics (McGraw‐Hill Book Company, Inc., New York, 1947).
15.
I am indebted to K. Takayanagi for the clear enunciation of this proof when in his recent review article on energy transfer (not yet published).
16.
D. Rapp, “Vibrational Energy Transfer in Quantum and Classical Mechanics,” LMSC Rept. 6‐90‐61‐14 (1961), Lockheed Missiles & Space Company, Palo Alto, California.
17.
K. Takayanagi, “Vibrational and Rotational Transitions in Molecular Collisions” (review paper to be published, Department of Physics, Saitama University, Japan).
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K.
Takayanagi
and T.
Kishimoto
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); , Prog. Theor. Phys.
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21.
K. Takayanagi and Y. Miamoto, Sci. Rept. Saitama University AIII, 101 (1959).
22.
R. E.
Turner
and D.
Rapp
, J. Chem. Phys.
35
, 1076
(1961
). The algebraic error contained in this paper is such that one should not introduce the factor but retain Thus only the modifications (i) and (iii) should be retained in that paper. Thus as the probability calculated with a Morse potential goes asymptotically to the probability with a simple exponential potential. The numerical changes in Table I are completely negligible.23.
R. A.
Allen
and P.
Feuer
, J. Chem. Phys.
40
, 2810
(1964
). (preceding paper).24.
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1963
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