Paper detail

Remarks on Quantum Probability Backflow

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&#39;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&#39;s constant goes to zero and probability backflow becomes impossible.