  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.

Quantum Physics