In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.

1.
W. H.
Miller
,
J. Chem. Phys.
54
,
5386
(
1971
).
2.
P. A. M. Dirac, Quantum Mechanics (Oxford U.P., London, 1958).
3.
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw‐Hill, New York, 1965).
4.
(a)
W. H.
Miller
,
J. Chem. Phys.
53
,
1949
(
1970
);
W. H.
Miller
, (b)
53
,
3578
(
1970
); ,
J. Chem. Phys.
W. H.
Miller
, (c)
54
,
5386
; ,
J. Chem. Phys.
W. H.
Miller
, (d)
55
,
3150
(
1971
). ,
J. Chem. Phys.
For a recent review and additional references, see W. H. Miller, Advances in Chemical Physics (Wiley, New York, 1974), Vol. XXV.
5.
(a)
R. A.
Marcus
,
Chem. Phys. Lett.
7
,
525
(
1970
);
(b)
J. Chem. Phys.
54
,
3965
(
1971
);
(C)
J. N. L.
Connor
and
R. A.
Marcus
,
J. Chem. Phys.
55
,
5636
(
1971
);
(d)
W. H.
Wong
and
R. A.
Marcus
,
J. Chem. Phys.
55
,
5663
(
1971
);
(e)
R. A.
Marcus
,
J. Chem. Phys.
56
,
311
(
1972
);
R. A.
Marcus
, (f)
56
,
3548
(
1972
); ,
J. Chem. Phys.
(g)
J.
Stine
and
R. A.
Marcus
,
Chem. Phys. Lett.
15
,
536
(
1972
);
(h)
R. A.
Marcus
,
J. Chem. Phys.
57
,
4903
(
1972
);
(i)
D. E.
Fitz
and
R. A.
Marcus
,
J. Chem. Phys.
59
,
4380
(
1973
).
6.
(a)
M. C.
Gutzwiller
,
J. Math. Phys.
8
,
1979
(
1967
);
M. C.
Gutzwiller
, (b)
10
,
1004
(
1969
); ,
J. Math. Phys.
M. C.
Gutzwiller
, (f)
11
,
1791
(
1970
); ,
J. Math. Phys.
M. C.
Gutzwiller
, (d)
12
,
343
(
1971
).,
J. Math. Phys.
7.
B. C.
Eu
,
J. Chem. Phys.
57
,
2531
(
1972
).
8.
See, for example, K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966).
9.
Of course, if ℏ is in some sense small, then d will generally tend to remain small, so that in the presence of nonharmonic potentials, small ℏ does play a role in keeping the Ehrenfest error term small.
10.
See, for example, R. P. Feynman and A. R. Hibbs, Ref. 3.
11.
W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
12.
H. Goldstein, Classical Mechanics (Addison‐Wesley, Reading, MA, 1950).
13.
As noted earlier, αt can add a pure phase factor to the wavefunction. Presumably this is a correction to the remaining phase. Its effect has been neglected for this discussion; in any explicit calculation involving Eqs. (2.11), the phase of αt is of course included.
14.
See, for example, R. G. Newton, Scattering Theory of Waves and Particles (McGraw‐Hill, New York, 1966).
15.
D.
Secrest
and
B. R.
Johnson
,
J. Chem. Phys.
45
,
4556
(
1966
).
16.
E. J. Heller (in preparation).
17.
(a)
E. A.
McCullough
and
R. E.
Wyatt
,
J. Chem. Phys.
54
,
3578
(
1971
);
E. A.
McCullough
and
R. E.
Wyatt
, (b)
55
,
3592
(
1971
).,
J. Chem. Phys.
18.
J. O. Hirschfelder, C. J. Goebel, and L. W. Bruch, Wisconsin Theoretical Chemistry Institute Report WIS‐TCl‐515, 1974.
19.
(a)
S. A.
Lebedeff
,
Phys. Rev.
165
,
1399
(
1968
);
S. A.
Lebedeff
, (b)
Phys. Rev. D
1
,
1583
(
1970
). The author wishes to thank Professor R. E. Wyatt for bringing this work to his attention.
20.
J. O. Hirschfelder, A. C. Christoph, and W. E. Palke, Wisconsin Theoretical Chemistry Institute Report WIS‐TC‐514, 1974.
21.
See, for example, E. Isaacson and H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).
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