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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Amplitude-free correlation function based on an algebra for coordinate transformation in semiclassical integrals

Kazuo Takatsuka

Published Jun 28, 2001• 19 citations
Publisher

Abstract

We present an algebra that facilitates a systematic coordinate transformation in semiclassical integrals such as those between the initial and final value representations. Applying this algebra to Maslov-type semiclassical wave packet theory [A. Inoue-Ushiyama and K. Takatsuka, Phys. Rev. A 59, 3256 (1999)], a semiclassical correlation function is extracted, which is free of the amplitude factor that suffers an exponential divergence in a chaotic system.

DOI
10.1103/physreve.64.016224

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Physical Review A•May 1, 1999•27 citations
We present an algebra that facilitates a systematic coordinate transformation in semiclassical integrals such as those between the initial and final value representations. Applying this algebra to Maslov-type semiclassical wave packet theory [A. Inoue-Ushiyama and K. Takatsuka, Phys. Rev. A 59, 3256 (1999)], a semiclassical correlation function is extracted, which is free of the amplitude factor that suffers an exponential divergence in a chaotic system.
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.
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.
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.
Initial-value phase-space integral representations for a time-dependent propagator are obtained in the coordinate and momentum representations. To do so we first derive nonuniform semiclassical propagators for the various representations, obtaining the global-time semiclassical phase indices (Maslov indices) due to caustics. Results include readily implementable general phase index formulas for any type of caustic, including cases where the Morse index theorem is inapplicable. The method of obtaining the indices is general and based simply on concatenating uniform short-time propagators which also gives rise to alternative path-integral forms. Initial-value integral representations are then derived by introducing a method of extending short-time initial-value propagator formulas to global times via a simple stationary-phase asympotoic-equivalence approach. The integrals reduce to the nonuniform semiclassical propagators within the stationary-phase approximation, are uniform about caustics, and have integrand phases which properly account for the global-time phases in terms of appropriate Maslov indices. The initial-value integrals are also consistently derived via a canonical mapping procedure on the coordinate-space path integral. Initial-value integrals for time-dependent wave-function propgation are also given. Evaluation of the initial-value integral expressions do not require trajectory root searches for propagation.
Based on Maslov's multidimensional semiclassical theory, a semiclassical Green function in a mixed position-momentum space is derived, which can be used to locally fix the semiclassical divergency in singular regions in configuration or momentum space. It is shown that when a semiclassical Green function is transformed from one representation to another, its form is invariant. The mixed representation includes both coordinate and momentum representations of semiclassical Green functions as two special cases.
A semiclassical theory based on the Maslov-type wave packet is reported. The present approximation constitutes a class of semiclassical theory that brings about a hierarchy of theoretical structures below or equivalent to the standard semiclassical approximation to the Feynman kernel. We first propose a semiclassical wave function; that is, what we call the action-decomposed function (ADF). An ADF is propagated in time semiclassically along a ``single action surface,'' which is characterized in terms of a given initial momentum, with the phase being proportional to the F2(q,p){F}_{2}(q,p)F2​(q,p)-type generating function. An arbitrary wave function can be expanded continuously in the ADF's of different initial momenta, each of which is associated with weighting factors both in configuration and momentum spaces. Depending on the form of the weighting function, the present scheme covers a number of different levels of approximation, ranging from a ``single'' ADF to a semiclassical kernel. Thus our theory provides a way of connecting the Maslov-type semiclassical wave packet to the WKB theory based on the Feynman kernel. It is generally concluded that the above kernel limit attains the highest accuracy possible within the present scheme, but it requires a large number of classical trajectories to represent it. On the other hand, a single ADF yields a little less accurate results but demands the fewest representing classical trajectories due to the narrowest momentum distribution. In fact, we show numerically that the autocorrelation function represented in terms of a single ADF, from which the energy spectrum is extracted with a Fourier transformation, can be calculated with drastically fewer classical trajectories without losing much accuracy. We also present a limitation in applying the ADF, and show an interesting symptom arising from a pathological use of the approximation. The present theory is quite promising for a spectral analysis of the vibrational states of relatively large molecules, if applied appropriately.