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The Journal of Chemical Physics

Analysis of the Pechukas description of mixed quantum-classical dynamics

J. Liam McWhirter

Published Mar 1, 1999• 5 citations
Publisher

Abstract

We review the Pechukas stationary phase analysis which yields the semiclassical reduced propagator. This propagator describes the dynamics of a mixed quantum-semiclassical system. In addition, we review a variant of this analysis, presented by us in an earlier article [J. Chem. Phys. 108, 5683 (1998)], which yields a semiclassical reduced propagator written in terms of connected and partially connected “classical” paths. These paths are obtained by solving the concatenation of several short time interval Pechukas equations. We argue and then demonstrate numerically that the “energy” along these paths is generally piecewise conserved: conserved across one short time interval, but not across several such intervals. In our review of these analyses, we relax the assumption made by Pechukas that the magnitude of the transition amplitude associated with the quantum subsystem varies much more slowly with changes in the classical subsystem’s trajectory than its phase. As our analyses demonstrate, this assumption serves to simplify the evaluation of the normalization path integral; the stationary phase paths are not affected by the making of this assumption. Solving the Pechukas equation subject to the initial configuration and velocity of the classical subsystem yields a collection of “classical” paths; the solution is nonunique. We provide a short time uniqueness theorem pertaining to the class of functional differential equations to which the Pechukas equation belongs; then, we review the sample problem that Pechukas used to first demonstrate this nonuniqueness. The theorem and the sample problem allow us to identify which parts within the Pechukas equation’s structure are responsible for the nonuniqueness in its solution. This nonuniqueness is verified numerically. Here, we show that the “energy” is conserved along each of the “classical” paths in the collection. However, the “energies” of any two paths in this collection will be the same only if the initial state specified in the Pechukas equation is an instantaneous adiabatic eigenstate of the quantum subsystem Hamiltonian which contains the interaction potential between the quantum and classical subsystems.

DOI
10.1063/1.478300

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Physical Review A•Jun 1, 1989•15 citations
We review the Pechukas stationary phase analysis which yields the semiclassical reduced propagator. This propagator describes the dynamics of a mixed quantum-semiclassical system. In addition, we review a variant of this analysis, presented by us in an earlier article [J. Chem. Phys. 108, 5683 (1998)], which yields a semiclassical reduced propagator written in terms of connected and partially connected “classical” paths. These paths are obtained by solving the concatenation of several short time interval Pechukas equations. We argue and then demonstrate numerically that the “energy” along these paths is generally piecewise conserved: conserved across one short time interval, but not across several such intervals. In our review of these analyses, we relax the assumption made by Pechukas that the magnitude of the transition amplitude associated with the quantum subsystem varies much more slowly with changes in the classical subsystem’s trajectory than its phase. As our analyses demonstrate, this assumption serves to simplify the evaluation of the normalization path integral; the stationary phase paths are not affected by the making of this assumption. Solving the Pechukas equation subject to the initial configuration and velocity of the classical subsystem yields a collection of “classical” paths; the solution is nonunique. We provide a short time uniqueness theorem pertaining to the class of functional differential equations to which the Pechukas equation belongs; then, we review the sample problem that Pechukas used to first demonstrate this nonuniqueness. The theorem and the sample problem allow us to identify which parts within the Pechukas equation’s structure are responsible for the nonuniqueness in its solution. This nonuniqueness is verified numerically. Here, we show that the “energy” is conserved along each of the “classical” paths in the collection. However, the “energies” of any two paths in this collection will be the same only if the initial state specified in the Pechukas equation is an instantaneous adiabatic eigenstate of the quantum subsystem Hamiltonian which contains the interaction potential between the quantum and classical subsystems.
We consider the time correlation function of observables pertaining to a (quantum subsystem +bath), where the bath is coupled to a reservoir with many degrees of freedom. Integrating over the coordinates of this reservoir and assuming no initial correlations between the (quantum subsystem+bath) and the reservoir, we obtain an expression for the time correlation function that contains an influence functional. We then take the semiclassical and Fokker–Planck limits while modeling the reservoir with an Ohmic continuum of harmonic oscillators coupled bilinearily to the coordinates of the bath. The semiclassical limit is taken using a variant of Pechukas’ stationary phase analysis of the reduced propagator that yields a time correlation function written in terms of connected “classical” paths. These paths are got by solving the concatenation of several short-time interval Pechukas equations; as a result, the determination of these paths is more feasible than the determination of the “classical” path associated with a single long-time interval Pechukas equation. This concatenation includes the dissipative and stochastic forces associated with a classical Brownian particle. We then use decoherence arguments derived from an inspection of the influence functional to eliminate the phase interference structure of the bath. This elimination yields a mixed quantum-classical time correlation function that can be evaluated using nonadiabatic mixed quantum-classical dynamics schemes similar to those proposed recently by Webster and Tully.
The extremal paths that arise in the stationary phase evaluation of coherent state path integrals do not seem to have a simple physical interpretation, in contrast to the extremal paths that occur in the conventional path integrals. On the other hand, a recently derived semiclassical formula for the coherent state propagator involves a path that is determined in exactly the same manner as the extremal paths of the conventional path integrals. Since both the semiclassical and the stationary phase analyses yield asymptotic (h(cross) to 0) approximations, the stationary phase and the semiclassical expressions for the propagator should be identical. The author presents a simple and direct proof that, in spite of the apparent differences, this is indeed the case. The simplification in the semiclassical formula is due to the utilisation of an appropriate set of canonical variables to describe the classical dynamics. In order to illustrate the usefulness of the semiclassical formula they present an application to the problem of the degenerate parametric amplifier, which had been treated before by operator ordering and path integral methods. The semiclassical approach has a simple classical interpretation that is absent in the alternative treatments.
AbstractThis work investigates a variational perspective on the Feynman path integral, aiming to explore connections between stationary action, probability amplitude, integrating over path evolutions, and the semiclassical limit. The variational analysis employed should be viewed as conceptually explorative and heuristic rather than strictly formal. It explores how variations of path evolutions in the path integral at the level of a single infinitesimal slice can yield meaningful insight into the behavior of quantum trajectories. A local variational analysis of the free particle and a particle near a constant potential provides insight into how the stationary-action principle underlies both classical and quantum behavior, through a local relationship between the classical slice-action and a functional over the phase-weighting of quantum path evolutions. In particular, this functional is proportional to the slice-action itself up to additive and multiplicative constants. This proportionality implies that the stationarity of this functional is equivalent to the stationarity of the slice-action, which in turn generates the semiclassical stationary path evolution. The resulting formulation produces results consistent with known short-time propagators for the free particle and a particle near a constant potential, including tunneling behavior.
In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.
Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system.\nA promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system.\nImposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom.\nAny numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros.\n\nIn this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators.\nThe result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian.\nThis equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived.\nCalculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.
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.