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Physical Review A

Geometrical evaluation of the Maslov index

Satoshi Takahashi, Kazuo Takatsuka

Published Feb 20, 2004• 10 citations
Publisher

Abstract

A geometrical method to calculate the Maslov index, which is an important part of the quantum phase, is proposed. This method is particularly useful for the recently proposed amplitude-free quasicorrelation function approach for the quantization of chaos [K. Hotta and K. Takatsuka, J. Phys. A 36, 4785 (2003)]. In this theory the Maslov index is involved with no need to calculate the amplitude factor, which is usually obtained through integration of the stability matrix. Since this matrix constitutes the major origin of difficulty in semiclassical quantization of chaos and since its integration is time consuming, the present method, which avoids the stability matrix, should assist in opening a gate for practical semiclassical quantization of chaos in a large molecular system.

DOI
10.1103/physreva.69.022110

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A geometrical method to calculate the Maslov index, which is an important part of the quantum phase, is proposed. This method is particularly useful for the recently proposed amplitude-free quasicorrelation function approach for the quantization of chaos [K. Hotta and K. Takatsuka, J. Phys. A 36, 4785 (2003)]. In this theory the Maslov index is involved with no need to calculate the amplitude factor, which is usually obtained through integration of the stability matrix. Since this matrix constitutes the major origin of difficulty in semiclassical quantization of chaos and since its integration is time consuming, the present method, which avoids the stability matrix, should assist in opening a gate for practical semiclassical quantization of chaos in a large molecular system.
In a recent paper [Takatsuka, Takahashi, Koh and Yamashita, J. Chem. Phys. 126 (2007), 021104], it was shown that the semiclassical quantization of chaos can be accurately achieved only with the phases (the action and Maslov phases) without use of the amplitude (preexponential) factor, which is known to diverge exponentially in chaos. The aim of this paper is to analyze the role of the semiclassical amplitude in energy quantization from various points of view, and we actually show that it is indeed quite limited. Therefore even chaos is quantized mainly by the phase constructive and destructive interferences.
A detailed framework for a new formalism of phase-space path integral is presented, the outline of the theory of which has been reported elsewhere [K. Takatsuka, Phys. Rev. Lett. 61, 503 (1988)]. This path integral is described in terms of a phase-space distribution function, which was proposed earlier by the present author under the name of the dynamical characteristic function (DCF). The DCF is characterized by two independent wave functions and with two phase spaces. In the present paper, we extend the DCF by utilizing the Feynman kernel in place of the two independent wave functions. This is called the identity DCF. The dynamics of the DCF for general wave packets is reduced to that of the identity DCF. Some characteristics of the identity DCF are presented. For example, the quantum-mechanical time-evolution operator is represented in terms of the identity DCF and of a complete set of quantum q (coordinate) and p (momentum) operators. A semiclassical theory for the identity DCF is also developed, its primary aim being the study of heavy-particle dynamics such as molecular collisions.It is shown that the propagation of the identity DCF is represented by classical trajectories, each of which is associated with an amplitude factor and with the action integral as the quantum phase factor. This amplitude factor is free from singularity, unlike the primitive semiclassical expression of the Feynman kernel, which is well known to diverge at caustics. The physical significance of the identity DCF is analyzed in the context of chaotic dynamics. In particular, it is shown that its amplitude factor is an explicit function of the eigenvalues of a Jacobian matrix [\ensuremath{\partial}(qf{\mathit{q}}_{\mathit{f}}qf​,pf{\mathit{p}}_{\mathit{f}}pf​)/\ensuremath{\partial}(qi{\mathit{q}}_{\mathit{i}}qi​,pi{\mathit{p}}_{\mathit{i}}pi​)], where (qi{\mathit{q}}_{\mathit{i}}qi​,p){\mathit{p}}_{)}p)​ and (qf{q}_{f}qf​,pf{p}_{f}pf​) are the initial and final points, respectively, of a classical trajectory. These eigenvalues are well known in the study of the classical chaos of Hamilton systems and have been used to construct a measure of the linear (and local) stability of an orbit, the Liapounov characteristic exponents, and so on. Further, if the identity DCF is applied to a periodic orbit, the amplitude factor becomes equivalent to the square root of the determinant of a matrix, which is the so-called monodromy matrix minus the unit matrix. This matrix is one of the standard quantities used in bifurcation theory to investigate the stability of fixed points. In fact, the amplitude factor is closely related to the so-called residue, which was proposed by Greene [J. Math. Phys. 20, 1183 (1979)] in order to predict the occurrence of global chaos in classical systems. It is emphasized that all these classical quantities arise naturally in our representation of quantum mechanics.
The mechanism of semiclassical quantization is examined in detail. We found an important role of destructive interference in building quantum spectrum. Based on this analysis, we propose an efficient method to calculate the semiclassical spectrum. This method is particularly promising in multidimensional chaotic systems and also very useful to identify which trajectories dominate a quantum spectrum. We numerically examine this method by applying to a two-dimensional chaotic system and show an illustrative example of identifying such a dominant trajectory that corresponds to an eigenstate.
Through semiclassical methods the subject of quantum chaos motivates and depends on Hamiltonian chaos research. Presented here is a selection of some of the most important Hamiltonian chaos topics that in this way get directly related to any of a variety of quantum chaos research problems. The overall emphasis is on intuitive explanations and illustrations of various ideas with the references containing more mathematically rigorous expositions. The chapter begins with a description of various useful theoretical and computational tools of chaos research, e.g.~surfaces of section, paradigms of chaos, stability analysis, and symbolic dynamics... This is followed by discussions regarding the geometry of chaos, how chaotic systems respond to perturbations, and the complexification of Hamiltonian dynamics. Most of the contents concern topics that are long established and well known by now. The exceptions involve complications that arise from the complexification of phase space variables and time. There are relatively recent results summarized regarding the appropriate branches of determinant phases and how to distinguish complex trajectories that are associated with classically allowed processes versus those associated with non-allowed ones.
This paper reports a method to build a semiclassical wave function corresponding to an invariant torus satisfying Einstein-Brillouin-Keller quantization conditions. Instead of calculating the stability matrix of the trajectory at each step, as in the standard method of Keller [Ann. Phys. (N.Y.) 4, 180 (1958)] or the modification of Maslov and Fedoriuk [Semiclassical Approximations in Quantum Mechanics (Reidel, Dordrecht, 1981)], we use the actual density of the trajectory, calculated by running the trajectory and counting passages through cells in coordinate space. The method is tested for a system of coupled Morse oscillators, and found to be comparable in accuracy to the standard method. It may be more useful than the standard method for testing ideas for semiclassical quantization in the chaotic regime.