Simple approach to computing tunneling time: test cases

Abstract

We suggest a recipe for calculating the time taken by a particle to tunnel out from an initially localized state in one of the wells of symmetrical double-well potential into the other well. We calculate average velocity v of the tunneling particle by solving the Schrodinger equation numerically for ψ(x, t), then estimating <v(t)> = (d/dt)<x(t)> and time-averaging it. The time taken to tunnel is measured by calculating T_av = l₀/v where l₀ is an idealized estimate of the barrier width. We suggest an approximate partitioning of tav into an intrinsic decay time (T_d) and a barrier crossing time (T_b), T_d being obtained from energy spread of the packet. T_av - td is shown to be close to the Wentzel-Brillouin-Kramers estimate of barrier crossing time tsc. The response of tav to an external driving field and nonzero temperatures are tested. We also apply the method to a purely barrier penetration problem as well as to the problem of tunneling through a fluctuating barrier.