Tunneling by an electron packet with an initially sharp wavefront

We have studied the time behavior of electron wavepackets traversing one-dimensional potentials. These packets are described as plane wave states that have been cut off to give a sharp initial wavefront. We find that the shift, or delay time, of the main part of the pulse is comparable to that of a Gaussian wavepacket with the same momentum, and that both kinds of pulses have the same broad-pulse (sharp-momentum) limit. This is shown to hold for a general class of potentials. We show explicitly that the steepest descents calculation described by Stevens does not lead to a finite tunneling velocity. An exact expression is given for the packet transmitted through a delta-function barrier, which suggests a new interpretation of the tunneling velocity that has been obtained in other calculations.