Wave Propagation in Linear Media | re-examined

4 One-dimensional quantum tunnelling
Thinking the results over again, we recognise that the aforementioned speculations have no
foundation. The wave packet seems to be advanced, but as a matter of fact, it is not. Rather,
we have one more example of pulse reshaping, an e ect we encountered already in section
2.1. It is the high-frequency components in the wave packet that arrive earlier at the barrier
than the rest and in addition have a greater tunnelling probability (or penetration depth). So
it is these spectral components that constitute the packet behind the barrier, which wehave
seen from the increased group velocity. Hence the peak is not conserved in the tunnelling
process. The fact that we could draw a trajectory inside the barrier stems from the de nition
of the maximum we decided to observe. Had we not taken the temporal maximum of the
wave function, but the spatial counterpart, we would not have been able to nd a peak within
the barrier at all, due to the exponential decay.
There is still another aspect of the trajectory interpretation: if the barrier is wide enough that
the e ects mentioned above are signi cant, the maximum of the probability density beyond
the wall becomes vanishingly small. So the trajectory covers the important fact that we are
trading propagation velocity for tunnelling probability or, in other words and from a practical
point of view, signal speed for signal energy.
Though being con ned to the evanescent region, the spectrum of the initial pulse in the
previous examples still had a considerable width originating from the fact that we investigated
very `short' electrons with k = 6. This is of course no reason to raise doubts about the
correctness and validity of the results, in particular the nding that pulse reshaping would
allow for negative tunnelling times, if the peak of the wave packet is taken as a reference point
for measuring the propagation. Yet it is sensible to consider the behaviour of electrons with
a narrower spectrum. We can expect that in such a case, the acceleration of the transmitted
wave packet will be less prominent because the spectrum rapidly decreases to both sides of
the original centre, leaving little room for a frequency shift. Consequently, the backward-shift
in time of the trajectory inside the barrier will occur to a lesser extent, so that eventually
the tunnelling time will reach a small but positive value. Indeed, g. 4.34 satis es these
expectations. The extrapolated trajectory of the incident wave packet is nearly parallel to
that of the transmitted wave, and the small velocity di erence is noticeable only in the
numerical determination of the trajectory ascent.
Let us now try to establish a connection to the various tunnelling time de nition of the last
section. Remaining with the trajectory approach, we must nd unambiguous de nitions of the
paths the wave packets move along. Behind the barrier, this is easy because the trajecory is
linear. Immediately in front of the barrier, however, interference impairs a clear perception of
the maximum. Obviously we must x the trajectory to points far away from the barrier where
the incident wave is still undisturbed. We have seen already that the velocity of the wave
packet can safely be assumed to be the group velocity of its carrier frequency, and we know
of course the exact initial position x0 of the peak at t =0. Wecan then calculate the time
the wave packet takes to arrive at the barrier | it is simply the propagation time required
to cover the distance x0 if no barrier is present at all. The moment when the transmitted
peak emanates from the rear interface can be computed from the trajectory like before, and
subtracting the two times we obtain what can be regarded as the tunnelling time. This is the
approach pursued also by Collins et al. [83].
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