Analytic Examples, Measurement Models and Classical Limit of Quantum Backflow

TL;DR

This paper analyzes the quantum backflow effect, providing analytical wave functions with significant backflow, and explores its classical limit and measurement implications, revealing a surprising parameter-independent quantum property.

Contribution

It introduces analytically computable wave functions exhibiting strong backflow and investigates the classical limit, highlighting a unique parameter-independent quantum effect.

Findings

Identified wave functions with up to 70% of maximum backflow

Discovered the backflow parameter $c_{bm}$ is independent of $hbar

Discussed measurement models for detecting backflow

Abstract

We investigate the backflow effect in elementary quantum mechanics - the phenomenon in which a state consisting entirely of positive momenta may have negative current and the probability flows in the opposite direction to the momentum. We compute the current and flux for states consisting of superpositions of gaussian wave packets. These are experimentally realizable but the amount of backflow is small. Inspired by the numerical results of Penz et al (M.Penz, G.Gr\"ubl, S.Kreidl and P.Wagner, J.Phys. A39, 423 (2006)), we find two non-trivial wave functions whose current at any time may be computed analytically and which have periods of significant backflow, in one case with a backwards flux equal to about 70 percent of the maximum possible backflow, a dimensionless number $c_{bm} \approx 0.04 $, discovered by Bracken and Melloy (A.J.Bracken and G.F.Melloy, J.Phys. A27, 2197 (1994)). This number has the unusual property of being independent of $\hbar$ (and also of all other parameters of the model), despite corresponding to an obviously quantum-mechanical effect, and we shed some light on this surprising property by considering the classical limit of backflow. We discuss some specific measurement models in which backflow may be identified in certain measurable probabilities.