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Progress of Theoretical Physics Supplement

WKB Theory of Tunneling between Tori

Shoji Takada, Hiroki Nakamura

Published Feb 1, 1994• 3 citations
Publisher

Abstract

General WKB theory of multidimensional tunneling between tori is formulated and illuminating physical picture on the effects of multidimensionality is provided. On the basis of Wilkinson's work, (1) Maslov's semiclassical wavefunction in classically accessible region is connected to the wavefunction in classically inaccessible region and (2) the latter is propagated into deep tunneling region. It is found that there exist two distinct regions of tunneling : a region where action is pure imaginary (I-region) and another region where action is complex (C-region). In I-region tunneling path can be defined by a certain classical trajectory on the inverted potential, while in C-region no tunneling path can be defined and the Huygens type wave propagation should be carried out. Moreover, wavefunction of a vibrationally excited state has nodal lines in C-region but not in I-region.

DOI
10.1143/ptps.116.295

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General WKB theory of multidimensional tunneling between tori is formulated and illuminating physical picture on the effects of multidimensionality is provided. On the basis of Wilkinson's work, (1) Maslov's semiclassical wavefunction in classically accessible region is connected to the wavefunction in classically inaccessible region and (2) the latter is propagated into deep tunneling region. It is found that there exist two distinct regions of tunneling : a region where action is pure imaginary (I-region) and another region where action is complex (C-region). In I-region tunneling path can be defined by a certain classical trajectory on the inverted potential, while in C-region no tunneling path can be defined and the Huygens type wave propagation should be carried out. Moreover, wavefunction of a vibrationally excited state has nodal lines in C-region but not in I-region.
A general Wentzel–Kramers–Brillouin (WKB) theory of multidimensional tunneling is formulated and an illuminating physical picture of the effects of multidimensionality is provided. Two basic problems are solved: (i) Maslov’s semiclassical wave function in the classically accessible region is connected to the wave function in the classically inaccessible region and (ii) the latter is propagated into the deep tunneling region. It is found that there exist two distinct types of tunneling: pure tunneling and mixed tunneling. The former is the usual one in which the tunneling path can be defined by a certain classical trajectory on the inverted potential and its associated action is pure imaginary. In the latter case, no tunneling path can be defined and the Huygens-type wave propagation should be carried out. In this case, tunneling is always accompanied by classical motion in the transversal direction and the associated action is complex. A general procedure is presented for the evaluation of energy splitting ΔE in the double well. Moreover, under the locally separable linear approximation, a simple and convenient formula for ΔE is derived and is confirmed to work well by comparison with the exact numerical calculations.
The WKB method is commonly used in semiclassical approximations to the wave function in both the classically allowed and the forbidden regions of a one-dimensional potential. In a multidimensional space, the method can be adapted to construct wave functions in an ``allowed'' region from classical trajectories or wave normals. However, in the ``forbidden'' region the WKB wave function is in general specified by two sets of wave fronts, the equiphase and the equiamplitude surfaces or equivalently by two sets of paths defined to be normal to these surfaces, respectively. We present a Huygens-type construction for obtaining these wave fronts and paths, which reveals that for non-normal incidence the paths are coupled to each other. The analysis enables us to answer some of the basic questions concerning tunneling in multidimensional nonseparable potentials. A special and important case occurs when the incident wave is normal to the turning surface. We show that for normal incidence the path equations are decoupled and are equivalent to Newton's equations of motion for the inverted potential and energy.
A semiclassical theory of multidimensional tunneling is formulated to calculate the tunneling wave function, energy splitting in a double well and decay rate constant from a metastable state. First, the tunneling wave function is calculated by analytic continuation of a quantized torus prepared in analytic forms using either the Chapman–Garrett–Miller method or the Birkhoff–Gustavson normal form method. For a weakly nonintegrable system, tunnelings are confirmed to be classified into two qualitatively different domains; pure tunneling in the I region and mixed tunneling in the C region. Semiclassical wave functions agree with quantum mechanical ones within a few percent both in classically allowed and tunneling regions. Breakdown of this simple picture is exemplified for cases of relatively strong couplings. Second, expressions of the tunneling energy splitting in a double well potential and the decay rate from a metastable state are derived. The wave function near a well formulated above is connected with the semiclassical Green’s function in the deep tunneling region. The latter is expressed by complex trajectories which start from the complex quantized torus. A preliminary numerical comparison with the quantum mechanical value is also done for the energy splitting.
We apply the quasilinearization method of Bellman and Kalaba [Quasilinearization and Nonlinear Boundary-Value Problems (Elsevier, New York, 1965)] to find approximate solutions for the multidimensional quantum tunneling for separable as well as nonseparable wave equations. By introducing the idea of the complex ``semiclassical trajectory'' which is valid for the motion over and under the barrier, and which, in the proper limit, reduces to the real classical trajectory in the allowed region, we obtain an eigenvalue equation for the characteristic wave numbers. This eigenvalue equation is similar to the corresponding equation obtained from the WKB approximation and yields complex eigenvalues with negative imaginary parts. When the barrier changes very rapidly as a function of the radial distance, we can replace the concept of the semiclassical trajectory, which may not be applicable in this case, by the concept of a complex ``quantum trajectory.'' The trajectory defined either way depends on a constant of integration, and by minimizing the action with respect to this constant we can obtain the minimum escape path. The case of two-dimensional tunneling is discussed as an example of this method.
As was pointed out in Section 3.1, there are some interesting features in multidimensional tunneling compared with the one-dimensional case [30,31,46]. These are the following: (1) There exist two types of tunneling, pure tunneling and mixed tunneling. (2) No tunneling path can be defined in the case of mixed tunneling. (3) Vibrational excitation can suppress tunneling because of mixed tunneling. In this section, these peculiar features are numerically demonstrated and analyzed [31] by taking tunneling splitting in symmetric double well potential as an example. Three typical two-dimensional model potentials are used.
The paper presents a new semiclassical theory of multidimensional tunneling and its application to the decay problem. A simple semiclassical expression for the decay rate constant is derived in terms of complex valued family of classical trajectories in the decay valley while the algebraic form of the family in the tunneling region is found by means of classical canonical perturbation theory. This provides a tool to analytically continue the classical trajectories into the decay valley, with a numerical illustration of such a continuation being done for a model 2D decay rate problem. The calculated results for the rate constant are found to be in good agreement with the exact ones for the high levels where the suggested perturbative treatment is applicable. It is also shown that the formulated theory can be directly compared with a previously proposed hopping method which gives a way to examine the accuracy of the latter without exact quantum calculations.