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.

21 pages, 2 figures. Different title, extended discussion. To appear in PRA

References in corpus (5)

• Complexified path integrals, exact saddles and supersymmetry
• Time-of-arrival correlations
• Time-of-arrival probabilities and quantum measurements: II Application to tunneling times
• Time-of-arrival probabilities and quantum measurements: III Decay of unstable states
• Real-time particle-detection probabilities in accelerated macroscopic detectors

Cited by in corpus (4)

• Time of arrival and localization of relativistic particles
• Quantum Information in Relativity: the Challenge of QFT Measurements
• Generalised escape paths for dynamical tunneling in QFT
• Pointer-based model for state reduction in momentum space