Time averaging of weak values - consequences for time-energy and coordinate-momentum uncertainty

TL;DR

This paper explores how averaging weak values over time affects quantum uncertainty relations, revealing new principles and limitations in predicting particle properties with greater precision than traditional bounds.

Contribution

It introduces a weak value time-energy uncertainty principle and demonstrates how time averaging can surpass standard uncertainty limits in quantum measurements.

Findings

Time averaging of weak values leads to a new time-energy uncertainty relation.

It is possible to predict a particle's momentum more accurately than the standard quantum limit.

Simulations on scattering from a square barrier illustrate these effects.

Abstract

Using the quantum transition path time probability distribution we show that time averaging of weak values leads to unexpected results. We prove a weak value time energy uncertainty principle and time energy commutation relation. We also find that time averaging allows one to predict in advance the momentum of a particle at a post selected point in space with accuracy greater than the limit of $\hbar /2$ as dictated by the uncertainty principle. This comes at a cost - it is not possible at the same time to predict when the particle will arrive at the post selected point. A specific example is provided for one dimensional scattering from a square barrier.