Effects of relativity on the time-resolved tunneling of electron wave packets

We solve numerically the time-dependent Dirac equation for a quantum wave packet tunneling through a potential barrier. We analyze the spatial probability distribution of the transmitted wave packet in the context of the possibility of effectively superluminal peak and front velocities of the electron during tunneling. Both the Dirac and Schrödinger theories predict superluminal tunneling speeds. However, in contrast to the Dirac theory the Schrödinger equation allows a possible violation of causality. Based on an analysis of the tunneling process in full temporal and spatial resolution, we introduce an instantaneous tunneling speed that can be computed inside the potential barrier.