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Progress of Theoretical Physics

Semiclassical Coherent State Path Integral around a Classically Realizable Trajectory

Atushi Tanaka

Published Sep 1, 2003• 0 citations

Abstract

The structure of the semiclassical Feynman kernel in the coherent state representation is studied.The analysis is focused on classically realizable (i.e.real-valued) trajectories, whose contributions to the semiclassical kernel persist in the limit → 0. A smooth deformation of the real-valued guiding trajectory that starts from the center of the "initial" coherent state and obeys the Hamilton equation induces complex-valued trajectories that contribute to the semiclassical kernel "linearized" around the guiding trajectory.The linearized semiclassical kernel describes Heller's Gaussian wavepacket dynamics (GWD).Two mechanisms that break the GWD approximation are studied in terms of the semiclassical coherent state path integral.The rate at which a caustic approaches the guiding trajectory is estimated.In addition, the deviation of the semiclassical kernel from a Gaussian shape is estimated, with the assumption that the influence of caustics is negligible.These results provide an estimation of departure times between classical and quantum evolutions.In particular, for classically integrable systems, the present argument elucidates that the departure time due to caustics is much larger than that due to the deviation from a Gaussian shape.Furthermore, a disagreement between the estimations of the departure times obtained from the Husimi representation and the Wigner representation is pointed out.

DOI: 10.1143/ptp.110.407

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We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

We consider a semiclassical approximation, first derived by Heller and coworkers, for the time evolution of an originally Gaussian wave packet in terms of complex trajectories. We also derive additional approximations replacing the complex trajectories by real ones. These yield three different semiclassical formulae involving different real trajectories. One of these formulae is Heller's thawed Gaussian approximation. The other approximations are non-Gaussian and may involve several trajectories determined by mixed initial–final conditions. These different formulae are tested for the cases of scattering by a hard wall, scattering by an attractive Gaussian potential and bound motion in a quartic oscillator. The formula with complex trajectories gives good results in all cases. The non-Gaussian approximations with real trajectories work well in some cases, whereas the thawed Gaussian works only in very simple situations.

The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V ( x ) = 1/ x 2 , where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.

The time-dependent WKB approximation for a coherent state is expanded to third order around a guiding real trajectory, allowing for the novel treatment of nonlinearity in its semiclassical dynamics, which is generally present in all physical systems far from the classical regime. The result is a closed-form solution consisting of a linear combination of Airy functions and their derivatives multiplied by an exponential. The expression's ability to capture nonlinearity is demonstrated by examining the autocorrelation of initial coherent states in anharmonic systems with few to many contributing periodic orbits. Its accuracy is compared to the quadratic expansion and found to be superior in regimes of ℏ where the curvature begins to be significant, as expected. Moreover, the expression is shown to be a real-trajectory uniformization over two coalescing saddle points that are emblematic of significant curvature. This extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time and establishes their regime of validity.

A semiclassical approximation to the time evolution of coherent states may be derived from a saddle point approximation to the exact quantum propagator, and in general it involves complex classical dynamics. We generalize previous one-dimensional results to

d

dimensions, and for the case

d=2

we present several applications. We also consider other simple approximations that depend only on real classical trajectories, but are not initial value representations. These approximations are able to reproduce interference and tunneling effects and involve propagating a few classical initial conditions compatible with the quantum uncertainties.