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Progress of Theoretical Physics

Semi-Classical Quantization for Multi-Dimensional Coupled-Channel Equation

Kazuhiro Yabana, H. Horiuchi

Published Mar 1, 1987• 10 citations
Publisher

Abstract

Multi-dimensional coupled-channel equation is treated by the WKB approximation for the main purpose of deriving the semi-classical quantization rule for it. A somewhat general type of the multi-dimensional coupled-channel Hamiltonian is presented which enables us to derive the semiclassical quantization rule for the energy spectra. As in the one-dimensional case, the WKB wave function has an additional phase besides the usual phase ∫Σpsdxs, and the quantization rule should include the contribution from this additional phase. The structure of the WKB wave function in the adiabatic representation is also discussed.

DOI
10.1143/ptp.77.517

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Multi-dimensional coupled-channel equation is treated by the WKB approximation for the main purpose of deriving the semi-classical quantization rule for it. A somewhat general type of the multi-dimensional coupled-channel Hamiltonian is presented which enables us to derive the semiclassical quantization rule for the energy spectra. As in the one-dimensional case, the WKB wave function has an additional phase besides the usual phase ∫Σpsdxs, and the quantization rule should include the contribution from this additional phase. The structure of the WKB wave function in the adiabatic representation is also discussed.
The extended WKB method recently presented is applied to the construction of a high-lying semiclassical wave function, corresponding to the quantum numbers (30,30), for the Barbanis Hamiltonian, a well-known two-dimensional nonseparable system; the results are compared with the semiclassical wave function obtained, for the same state, by Davis and Heller, by means of the coherent Gaussians method. The extended WKB method shows that the semiclassical wave function is the sum of four wave functions, differing on parts of the classically allowed region, each of these separately satisfying the Einstein–Brillouin–Keller quantization rules. The results presented here show that our method, which is the direct extension to integrable nonseparable Hamiltonians of the usual WKB scheme, is suitable for the semiclassical quantization of high-lying states also, and enables a closer investigation of the fine structures of the semiclassical wave functions.
While the scattering phase for several one-dimensional potentials can be exactly derived, less is known in multi-dimensional quantum systems. This work provides a method to extend the one-dimensional phase knowledge to multi-dimensional quantization rules. The extension is illustrated in the example of Bogomolny's transfer operator method applied in two quantum wells bounded by step potentials of different heights. This generalized semiclassical method accurately determines the energy spectrum of the systems, which indicates the substantial role of the proposed phase correction. Theoretically, the result can be extended to other semiclassical methods, such as Gutzwiller trace formula, dynamical zeta functions, and semiclassical Landauer–Büttiker formula. In practice, this recipe enhances the applicability of semiclassical methods to multi-dimensional quantum systems bounded by general soft potentials.
With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.
An algebraic reformulation of the Bohr - Sommerfeld (BS) quantization rule is suggested and applied to the study of bound states in one-dimensional quantum wells. The energies obtained with the present quantization rule are compared to those obtained with the usual BS and WKB quantization rules and with the exact solution of the Schrödinger equation. We find that, in diverse cases of physical interest in molecular physics, the present quantization rule not only yields a good approximation to the exact solution of the Schrödinger equation, but yields more precise energies than those obtained with the usual BS and/or WKB quantization rules. Among the examples considered numerically are the Poeschl - Teller potential and several anharmonic oscillator potentials, which simulate molecular vibrational spectra and the problem of an isolated quantum well structure subject to an external electric field.