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

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