Wave packet propagation across barriers by semiclassical initial value methods

Abstract

Semiclassical initial value representation (IVR) formulas for the propagator have difficulty describing tunneling through barriers. A key reason is that these formulas do not automatically reduce, in the classical limit, to the version of the Van Vleck-Gutzwiller (VVG) propagator required to treat barrier tunneling, which involves trajectories that have complex initial conditions and that follow paths in complex time. In this work, a simple IVR expression, that has the correct tunneling form in the classical limit, is derived for the propagator in the case of one-dimensional barrier transmission. Similarly, an IVR formula, that reduces to the Generalized Gaussian Wave Packet Dynamics (GGWPD) expression [D. Huber, E. J. Heller, and R. Littlejohn, J. Chem. Phys. 89, 2003 (1988)] in the classical limit, is derived for the transmitted wave packet. Uniform semiclassical versions of the IVR formulas are presented and simplified expressions in terms of real trajectories and WKB penetration factors are described. Numerical tests show that the uniform IVR treatment gives good results for wave packet transmission through the Eckart and Gaussian barriers in all cases examined. In contrast, even when applied with the proper complex trajectories, the VVG and GGWPD treatments are inaccurate when the mean energy of the wave packet is near the classical transmission threshold. The IVR expressions for the propagator and wave packet are cast as contour integrals in the complex space of initial conditions and these are generalized to potentially allow treatment of a larger variety of systems. A steepest descent analysis of the contour integral formula for the wave packet in the present cases confirms its relationship to the GGWPD method, verifies its semiclassical validity, and explains results of numerical calculations.

Wavepacket revivals via complex trajectory propagation

“…References [6,19,20] showed that complex time propagation, along with the correct choice of complex time contours, leads to accurate wavepacket reconstruction for barrier transmission problems. Similarly, the oscillatory system studied in the following section is strongly influenced by the dynamical singularities, and the use of complex time integration contours is required to obtain accurate results.…”

Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics

“…This not only implies the existence of randomness in a quantum system, but also discloses that the random motion occurs in complex space. Numerous studies with the complex initial condition and the random property have been discussed [23][24][25]. A stochastic interpretation of quantum mechanics is proposed which regards the random motion as a nature property of the quantum world not the interference made by the measurement devices [26,27].…”

Communication: Systematic elimination of Stokes divergences emanating from complex phase space caustics

“…14,20,24 A separate publication will give more details of the implementation and the phase space geometry of the caustics in the four prototypical potentials.…”

“…However the PED requires comparison of trajectory pairs which can only be found numerically via a root search in the complex trajectory manifold. In continuous time dynamics such a root search is quite challenging and has been successful so far only for relatively short times [5,[11][12][13][14]. Long time dynamics where the trajectory manifold is plagued by a multitude of caustics, remains elusive.…”

“…In continuous time dynamics, such a root search is quite challenging and has been successful so far only for relatively short times. 5,[11][12][13][14] Long time dynamics where the trajectory manifold is plagued by a multitude of caustics, remains elusive.…”