Tunneling and the semiclassical propagator: A new perspective

Abstract

We consider quantum propagation in one dimension when there is an intervening barrier, from both the time domain and the energy domain viewpoints. We relate the issue of tunneling in the energy representation to flow across the barrier in real time. We find that time domain trajectories which have energies above the barrier do not correctly account for (energy domain) tunneling below the barrier when the time-energy Fourier transform is done exactly. The stationary phase transform (WKB result) cannot be thought of as an approximation to the Fourier integral, because of a branch cut in the complex-time plane. In fact, in the time domain, to obtain accurate semiclassical amplitudes for cross barrier flow at long times, below barrier tunneling trajectories need to be included. The Fourier transform from the time domain into the energy domain cannot be “improved” by performing it numerically rather than by stationary phase; instead, it is the time Green function kernel of the integral which needs correction (by the addition of tunneling paths) if one keeps time real.