Remarks on Quantum Probability Backflow

Abstract

It is known that for a non-relativistic quantum particle traveling freely on the $x$-axis, the positional probability can flow in the opposite direction to the particle's velocity. The maximum possible amount of such backflow that can occur over any time interval has been determined previously as the largest positive eigenvalue of a certain hermitian observable, with the value $0.0384517\dots$, or about $4\%$ of the total probability on the line. The eigenvalue problem is now considered numerically in the more general case of states with momentum restricted to the range $p_0<p<\infty$, for any given value $p_0$. It is found that the maximum possible backflow decreases monotonically, but never reaches $0$, as $p_0$ increases through positive values; and it increases monotonically, but never reaches $1$, as $p_0$ decreases through negative values. Both of these effects are non-classical. The results allow a simple interpretation of the classical limit, as an effective value of Planck's constant goes to zero and probability backflow becomes impossible.