Dwell time and phase times for transmission and reflection

The dwell time ${\tau }_D$(E) and the transmission and reflection times ${\tau }_T$(E) and ${\tau }_R$(E), respectively, associated with the interaction of a particle of energy E with a one-dimensional barrier are currently the subject of considerable controversy. In this paper we prove that, despite a recent claim to the contrary, Büttiker’s expression for the dwell time is correct. We use this fact in combination with a rigorous result derived by Hauge, Falck, and Fjeldly to prove that the phase times ${\tau }_T^{\phi }$(E) and ${\tau }_R^{\phi }$(E) cannot be identified with ${\tau }_T$(E) and ${\tau }_R$(E).