Quantum perturbation theory is used to examine the eigenvalues of a nonseparable Hamiltonian system in the classically regular and irregular regimes. As a function of the perturbation parameter, the eigenvalues obtained by exact (matrix diagonalization) methods undergo an avoided crossing. In the present paper perturbation theory is used as an approximate method to predict the locations of such avoided crossings in energy‐parameter space. The sparsity of such avoided crossings in the Hénon–Heiles system is seen to produce regular sequences in the eigenvalues even when the classical motion is predominantly chaotic.
REFERENCES
1.
See, e.g., P. J. Robinson and K. A. Holbrook, Unimolecular Reactions (Wiley, New York, 1972);
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A good collection of the relevant literature in this field may be found in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Volta Memorial Conference, Como, 1977, edited by G. Casati and J. Ford, Springer‐Verlag Lecture Notes in Physics, 93 (1979);
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(a) V. L. Arnol’d and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968);
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References to recent work are listed in
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(1980
).9.
(a) R. A. Marcus, Extended Abstracts, 27th Annual Conference on Mass Spectrometry and Allied Topics, Seattle, Wash., June 3–8, 1979;
(b) R. A. Marcus, in Horizons in Quantum Chemistry, Proceedings of the 3rd International Congress on Quantum Chemistry, Kyoto, 1979, edited by K. Fukui and B. Pullman (Reidel, Dordrecht Neth., 1980), p. 107;
(c) R. A. Marcus, in Nonlinear Dynamics, Conference Proceedings, N.Y.A.S., New York, 1980.
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G. E. O. Giacaglia, Perturbation Methods in Nonlinear Systems (Springer, New York, 1972).
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A. Dalgarno, in Quantum Theory, edited by D. R. Bates (Academic, New York, 1961).
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15.
16.
17.
The quantity is simply computed by summing the multiplicities of the zeroth order states of energy less than
18.
The crossing of 12, and 11, occurs at smaller λ; the avoided crossing of 13, and 12, was seen in Ref. 13.
19.
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© 1981 American Institute of Physics.
1981
American Institute of Physics
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