WKB approximation in complex time

Pedro Pereyra Padilla

K. T. Hecht

Yair Goldfarb, Jeremy Schiff, David J. Tannor

We present a significant improvement to a complex time-dependent WKB (CWKB) formulation developed by Boiron and Lombardi [J. Chem. Phys. 108, 3431 (1998)] in which the time-dependent WKB equations are solved along classical trajectories that propagate in complex space. Boiron and Lombardi showed that the method gives very good agreement with the exact quantum mechanical result as long as the wavefunction does not exhibit interference effects such as oscillations and nodes. In this paper, we show that this limitation can be overcome by superposing the contributions of crossing trajectories. Secondly, we demonstrate that the approximation improves when incorporating higher order terms in the expansion. Thirdly, equations of motion for caustics and Stokes lines are implemented to help overcome Stokes discontinuities. These improvements could make the CWKB formulation a competitive alternative to current time-dependent semiclassical methods.

R. Krivec, Victor B. Mandelzweig, Frank Tabakin A

Adrian Patrascioiu

It is shown that, in general, full agreement between the semiclassical approximation of the path integral and the WKB method of ordinary quantum mechanics is achieved only by the use of complex-time classical solutions. This technique is applied to compute the shift in energy levels due to tunneling and to calculate the lifetimes of virtual states. It is emphasized that the exponential time decay of the kernel is only approximately valid for finite time and that for $T\ensuremath{\rightarrow}\ensuremath{\infty}$ a power-law behavior applies. The Gaussian approximation of the kernel corroborates these statements.