A general barrier-traversal-time operator is constructed using the time-of-arrival formalism. We study the operator's dynamics and determine the role played by the time-energy commutation relation. It turns out that similar dynamics is observed whether the traversal-time operator is canonically conjugate to the system Hamiltonian or not. We also use the barrier-traversal-time operator to calculate the traversal-time distributions for different cases (free case, and above-the-barrier and under-the-barrier traversals). The peak of the traversal-time distributions coincides with the classical expected traversal times for the free case and the above-the-barrier case. We then present our interpretation of the time-energy uncertainty relation that is consistent with the different traversal-time distributions.

• Received 4 March 2018
• Revised 29 May 2018

DOI:https://doi.org/10.1103/PhysRevA.97.062127