Remarks on Quantum Probability Backflow

TL;DR

This paper investigates quantum probability backflow for particles with restricted momentum, showing how the maximum backflow diminishes or increases with momentum range, illuminating the classical limit of quantum behavior.

Contribution

It extends the analysis of quantum backflow eigenvalues to states with restricted momentum, revealing how maximum backflow varies with momentum bounds and clarifies the classical limit.

Findings

Maximum backflow decreases monotonically with increasing positive momentum cutoff.

Maximum backflow increases monotonically with decreasing negative momentum cutoff.

Results provide a simple interpretation of the classical limit as Planck's constant approaches zero.

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.