Does the Hartman effect exist in triangular barriers

Abstract

We study the phase, Larmor and dwell times of a particle scattered off triangular barriers (TBs). It is interesting that the dependences of dwell, reflective phase and Larmor times on the wave number, barrier width and height for a pair of mirror-symmetric (MS) exact triangular barriers (ETBs) are quite different, as the two ETBs have quite distinct scattering surfaces. In comparison, the dependence of the transmitted phase or Larmor times is exactly the same, since the transmitted amplitudes are the same for a pair of MS TBs. We further study the Hartman effect by defining the phase and Larmor velocities associated with the phase and Larmor times. We find no barrier width saturation effect for the transmitted and reflected times. This is indicated by the fact that all the velocities approach finite constants that are much smaller than the speed of light in vacuum for TBs with positive-slope impact faces. As for ETBs with vertical left edges, the naive velocities seem to also indicate the absence of the Hartman effect. These are quite distinct from rectangular barriers and may shed new light on the clarification of the tunneling time issues.