Relation between quantum tunneling times for relativistic particles

The time it takes for a particle or wave packet to tunnel through a potential barrier has been debated for decades [1–8]. The fact that there is a finite duration for the tunneling process is not in doubt. The debate centers around the validity of the various proposed tunneling times, the relation between those times, and the physical meaning of a tunneling time especially when it predicts apparent superluminal velocities. Two of the more commonly used tunneling times are the dwell time (defined through the integrated probability density under the barrier) and the phase time (defined by the energy derivative of the transmission phase shift). These two times, however, are not entirely unrelated. They are equal under certain circumstances but generally differ as a result of quantum interference effects [9–11]. Both times also saturate with increasing barrier length, a phenomenon known as the Hartman effect [2] and which has been claimed to lead to infinite tunneling velocities for opaque barriers [6]. This effect has recently been explained as arising from the saturation of stored energy or number of particles under the barrier [11–13]. With some exceptions [14–23], most discussions of quantum tunneling time have been based on the nonrelativistic Schrödinger equation even when apparent “faster than c” effects are considered. In particular there has been no discussion of the relation between the various tunneling times for relativistic particles. Here we derive an exact relation between the phase time and the dwell time for relativistic particles that satisfy the Dirac equation. We show by means of an explicit stationary state calculation that the phase time is equal to the dwell time plus a self-interference delay which is a relativistic generalization of earlier results. II. RELATIVISTIC ONE-DIMENSIONAL SCATTERING RELATIONS