Physical Mechanism for Apparent Superluminality in Barrier Tunneling

In quantum tunneling, a particle lacking the energy to go over a classically impenetrable potential barrier can nevertheless end up on the other side, albeit with small probability. This process is the basis for devices such as the scanning tunneling microscope, which enables imaging with atomic-scale resolution, and the resonant tunneling diode, used in fast electronic switches. An analogous phenomenon occurs in the propagation of electromagnetic waves which can tunnel as evanescent waves through forbidden regions.

For decades, physicists have struggled to answer the question “how long does it take for a particle or wave packet to tunnel through a barrier?” Theoretical calculations of the group delay, apparently confirmed by electromagnetic experiments, have been widely interpreted to mean that the tunneling velocity is superluminal, or faster than light. Furthermore, theory and experiment show that the group delay in tunneling saturates with barrier length, a paradoxical result known as the Hartman Effect.

In this talk we calculate the group delay for various barriers, relate it to the stored energy or integrated probability density in the barrier, and then resolve the mystery of apparent superluminality and the Hartman Effect in barrier tunneling.