Path of a tunneling particle

Abstract

Attempts to find a quantum-to-classical correspondence in a classically forbidden region leads to nonphysical paths involving, for example, complex time or spatial coordinates. Here, we identify genuine quasiclassical paths for tunneling in terms of probabilistic correlations in sequential time-of-arrival measurements. In particular, we construct the postselected probability density ${P}_{\text{p.s.}}(x,\ensuremath{\tau})$ for a particle to be found at time $\ensuremath{\tau}$ in position $x$ inside the forbidden region, provided that it later crossed the barrier. The classical paths follow from the maximization of the probability density with respect to $\ensuremath{\tau}$. For almost monochromatic initial states, the paths correspond to the maxima of the modulus square of the wave function ${|\ensuremath{\psi}(x,\ensuremath{\tau})|}^{2}$ with respect to $\ensuremath{\tau}$ and for constant $x$ inside the barrier region. The derived paths are expressed in terms of classical equations, but they have no analogs in classical mechanics. Finally, we evaluate the paths explicitly for the case of a square potential barrier.