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Physical Review Letters

Global Representation of Maslov-Type Semiclassical Wave Function and Its Spectrum in a Small Number of Classical Trajectories

Kazuo Takatsuka, Atsuko Inoue

Published Feb 24, 1997• 38 citations
Publisher

Abstract

An explicit solution to the Maslov-type semiclassical theory for propagating a wave function, rather than evolving in time the Feynman kernel, is presented. It turns out that the present solution bears distinguished advantages over the semiclassical kernel, one of the most remarkable examples of which is the far less number of classical trajectories required for the propagation, basically proportional to ${P}^{2N}\ensuremath{\sim}{P}^{3N}$ for the kernel while only to ${P}^{N}$ in our solution, where N is the dimension in configuration space and P is the number of sampling points in each dimension. As an illustrative example to show the validity of the solution, the theory is applied to the calculation of eigenvalues of the Morse oscillators, giving accurate results in a compact way.

DOI
10.1103/physrevlett.78.1404

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An explicit solution to the Maslov-type semiclassical theory for propagating a wave function, rather than evolving in time the Feynman kernel, is presented. It turns out that the present solution bears distinguished advantages over the semiclassical kernel, one of the most remarkable examples of which is the far less number of classical trajectories required for the propagation, basically proportional to P2N\ensuremath∼P3N{P}^{2N}\ensuremath{\sim}{P}^{3N}P2N\ensuremath∼P3N for the kernel while only to PN{P}^{N}PN in our solution, where N is the dimension in configuration space and P is the number of sampling points in each dimension. As an illustrative example to show the validity of the solution, the theory is applied to the calculation of eigenvalues of the Morse oscillators, giving accurate results in a compact way.
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.
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.
Rather general expressions are derived which represent the semiclassical time-dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time-dependent wave functions without searching for special trajectories that satisfy two-time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.
A new technique is developed to generate semiclassical wave functions. The method uses only information already available from a standard semiclassical quantization of a system. Linear superpositions of Gaussian coherent states that lie along quantizing classical trajectories are used, with phases given by the action integrals plus a Maslov-type correction. Wave functions generated in this way suffer from none of the problems with caustics that primitive semiclassical wave functions encounter. The semiclassical wave functions are convenient for subsequent use in applications, e.g., molecular spectra. By generating wave functions for several simple systems, we show that under most circumstances these wave functions are very accurate approximations to the true quantum states.
Employing the path integral formalism, we study a quantum (test) system coupled to an environment consisting of infinitely many harmonic oscillators. A simple semiclassical approxi- mation, in which only real classical trajectories are required, is used to derive the propagator of the reduced density matrix for Ohmic dissipation and high temperatures. The inclusion of a summation over trajectories with the correct Maslov phases in the final expression for the propagator permits investigation of nonharmonic test systems. In a numerical example for a Morse oscillator interacting with a high temperature heat bath, the semiclassical result, obtained by the method described here, correctly displays the suppression of quantum behavior as observed in the exact calculation.
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.