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arXiv:0908.1875

An Initial Value Representation with Complex Trajectories

Marcus A. M. de Aguiar, S. A. Vitiello, Adriano Grigolo

Published Aug 13, 2009• 0 citations
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Abstract

We present an Initial Value Representation for the semiclassical coherent state propagator based on complex trajectories. We map the complex phase space into a real phase space with twice as many dimensions and use a simple procedure to automatically eliminate non-contributing trajectories. The resulting semiclassical formulas do not show divergences due to caustics and provide accurate results.

DOI
10.48550/arxiv.0908.1875

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We present an Initial Value Representation for the semiclassical coherent state propagator based on complex trajectories. We map the complex phase space into a real phase space with twice as many dimensions and use a simple procedure to automatically eliminate non-contributing trajectories. The resulting semiclassical formulas do not show divergences due to caustics and provide accurate results.
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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 ddd dimensions, and for the case d=2d=2d=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.
Semiclassical time-dependent solutions to the Schroedinger equation can be obtained by use of integral formulas known as initial value representations (IVRs). Over the last few decades IVR formulas have established a sucessful reputation, most notably in the field of physical chemistry, for it is often found that the quantum behavior of molecular systems is very well approximated by semiclassical wavefunctions. The IVR input consists in a swarm of independent classical trajectories, each of which obeys the dynamics dictated by the system’s unquantized Hamiltonian. In a recent work [Aguiar et al, Chem. Phys. 370 (2010)] an IVR expression based on complex classical trajectories was formulated. The complex nature of the classical trajectories in this particular formula is a consequence of employing minimum uncertainty wavepackets (coherent states) to represent the time evolution operator upon which the formula is constructed. Here, an improved and simpler version of this complex IVR formula is presented. In order to test its accuracy, we apply it to a simple model system.
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A semiclassical formula for the coherent-state propagator requires the determination of specific classical paths inhabiting a complex phase-space and governed by a Hamiltonian flux. Such trajectories are constrained to special boundary conditions which render their determination difficult by common methods. In this paper we present a new method based on Runge-Kutta integrator for a quick, easy and accurate determination of these trajectories. Using nonlinear one dimensional systems we show that the semiclassical formula is highly accurate as compared to its exact counterpart. Further, we clarify how the phase of the semiclassical approximation is correctly retrieved during the time evolution.
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