Wave Propagation in Linear Media | re-examined

4.6 Examples of tunnelling events
the interaction took place. Seemingly, the barrier is too thin for a fully developed wave packet
to evolve. As we could have expected, the transmitted wave grows stronger as the barrier size
is reduced, and for the thin obstacle even surpasses the re ected wave in magnitude. Not at
all unexpected comes the form of the trajectories of the two wave packets ( gs. 4.26 and 4.27,
respectively). In the example of the medium-sized obstacle, the trajectory of the transmitted
wave packet is still more inclined than that of the impinging wave, albeit not to such an extent
as with the thick barrier above. For the thin barrier, on the other hand, this e ect is hardly
noticeable at all, which con rms the intuitive suspicion that a thin barrier does less harm to
the spectrum of a pulse since the evanescent components have a better chance of tunnelling
through.
Apart from the di erent inclinations of the trajectories, there is another instructive detail
the pictures reveal. Suppose we regard only the interfaces of the barrier in search of a
plausible way to de ne a tunnelling time. Intuitively, we could measure the delay between
the appearance of the wave packet's peak at the front and rear edge. Such an approach is,
however, deceptive in that it disregards the way the peak at the front interface is generated.
In fact, it is not the peak of the incident wave packet but it originates rather incidentally
from the superposition of incoming and re ected wave. Actually, it reaches the barrier later
than the free wave packet would without the presence of the obstacle. For the spectator at
the front edge of the barrier, it looks as though the approach of the packet is slowed down by
interference. Therefore the trajectory, if at all, is meaningful only in an asymptotic sense if
extrapolated from the undisturbed region to the barrier.
We now explore a di erent initial wave packet, namely one with a spectrum su ciently
con ned to the evanescent region. Since we can expect genuine tunnelling this time, we must
choose a rather thin barrier in order to obtain a recognisable wave packet behind the tunnel.
Indeed, g. 4.28 shows that a medium-sized obstacle already re ects almost the complete
incident wave, and practically nothing reaches the far side of the barrier. The situation is
naturally better for a thin barrier, as g. 4.29 demonstrates. It is clear, though, that the lower
the centre frequency of the wave is, the narrower the barrier must be in order to obtain a
noticeable tunnel e ect.
The inspection of the trajectories of the wave packets outside the barrier also yields nothing
unexpected. Like before, the path of the wave leaving the far side of the barrier is more
inclined towards the axis, which emphasises once more the high-pass behaviour of the barrier.
Consequently, this e ect is more pronounced if the barrier is wide. So far, the results qualitatively
agree with those obtained for the higher-frequency wave packet treated before. There
is, however, a signi cant di erence: from a comparatively thick barrier (like in g. 4.30), the
peak of the tunnelling wave packet emanates before the peak of a freely moving packet would
have reached the front of the barrier. Accordingly, the trajectory even runs backwards in time
inside the barrier. This phenomenon is still more impressive for thicker barriers ( gs. 4.32 and
4.33). At the rst super cial examination, this could be taken for a violation of causality,
because it indeed looks like turning the clock back while the electron is tunnelling through
the wall. Taking one step further, we could as well see this result as a proof for a wave packet
travelling faster than the speed of light | although the velocity of light actually never showed
up throughout the entire derivation of the tunnel e ect.
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