Path of a tunneling particle

Abstract: Attempts to find a quantum-to-classical correspondence in a classically forbidden region leads to non-physical paths, involving, for example, complex time or spatial coordinates. Here, we identify genuine quasi-classical paths for tunneling in terms of probabilistic correlations in sequential time-of-arrival measurements. In particular, we construct the post-selected probability density $P_{p.s.}(x, \tau)$ for a particle to be found at time $\tau$ 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 $\tau$. For almost monochromatic initial states, the paths correspond to the maxima of the modulus square of the wave-function $|\psi(x,\tau)|^2$ with respect to $\tau$ and for constant $x$ inside the barrier region. The derived paths are expressed in terms of classical equations, but they have no analogues in classical mechanics. Finally, we evaluate the paths explicitly for the case of a square potential barrier.