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Physical Review A

Semiclassical quantization via adiabatic switching. II. Choice of tori and initial conditions for multidimensional systems

Subhash Saini, Jakub Zakrzewski, Howard S. Taylor

Published Oct 1, 1988• 16 citations
Publisher

Abstract

The formulation of a general strategy for implementing semiclassical quantization using the adiabatic-switching method, presented in the preceding paper for two-dimensional systems, is generalized to multidimensional systems. The method is shown to produce accurate semiclassical eigenvalues for the realistic polynomial potential describing ${\mathrm{SO}}_{2}$ as well as for model potentials for other multidimensional systems. The results are compared with semiclassical results obtained with use of other methods.

DOI
10.1103/physreva.38.3900

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The formulation of a general strategy for implementing semiclassical quantization using the adiabatic-switching method, presented in the preceding paper for two-dimensional systems, is generalized to multidimensional systems. The method is shown to produce accurate semiclassical eigenvalues for the realistic polynomial potential describing SO2{\mathrm{SO}}_{2}SO2​ as well as for model potentials for other multidimensional systems. The results are compared with semiclassical results obtained with use of other methods.
The globally uniform semiclassical approximation for energy eigenstates developed by D. Zor and K. G. Kay [Phys. Rev. Lett. 76, 1990 (1996)] is derived explicitly for the case of multidimensional systems and is applied to two test cases. The adiabatic switching approximation is used to obtain various quantities that enter the semiclassical expression. Two formulations of the method are examined: one requiring several trajectories for each desired state and another requiring only a single trajectory per state. The multitrajectory version yields accurate results for all states investigated (overlap between semiclassical and quantum eigenstates >0.98), including some influenced by classical chaos. The single-trajectory treatment, however, is more efficient, gives accurate results for regular states, and is even applicable for certain chaotic states, although the multiple-trajectory method is preferred in such cases. Despite the substantial resemblance of the present theory to the frozen Gaussian approximation (FGA), it is a true semiclassical approximation and is found to produce wave functions that are significantly more accurate than those obtained from the FGA for all states examined.
The semiclassical method based on the Einstein-Brillouin-Keller quantization of invariant phase-space tori and the hypothesis of adiabatic invariance of classical action variables is applied in the calculation of highly excited energy levels of two-dimensional coupled oscillator systems. The results obtained are in good agreement with quantum-mechanical calculations even in the regions where the corresponding classical dynamics becomes apparently irregular. General limitations of the method are discussed.
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The problem of semiclassical quantization of nonseparable systems with a finite number of degrees of freedom is studied within the framework of Heisenberg matrix mechanics, in extension of previous work on one-dimensional systems. The relationship between the quantum theory and multiply-periodic classical motions is derived anew. A suitably averaged Lagrangian provides a variational basis not only for the Fourier components of the semiclassical equations of motion, but also for the general definition of action variables. A Legendre transformation to the Hamiltonian verifies that these have been properly chosen and therefore provide a basis for the quantization of nonseparable systems. The problem of connection formulas is discussed by a method integral to the present approach. The action variables are shown to be adiabatic invariants of the classical system. An elementary application of the method is given. The methods of this paper are applicable to nondegenerate systems only.