Wave Propagation in Linear Media | re-examined

4 One-dimensional quantum tunnelling
the tunnel and bounces back again, a fraction of it seems to get caught in the barrier, running
back and forth between the interfaces while decaying not too quickly. This peculiar e ect
has nothing to do with evanescent waves but stems from the ordinarily transmitted parts
of the spectrum. It is just the common observation made for propagating waves that every
change in the medium inevitably causes re ections. Looking carefully, we even discover a
small after-runner of the re ected wave (at T = 40 and X ,20 in the three-dimensional
plot), which can be traced back to the instant when the wave inside the barrier hits the left
interface (at X = 0) for the rst time.
To investigate this behaviour in greater detail, we regard an even thicker barrier and the same
electron as before (as in the close-up of g. 4.22). In this case, the bouncing of the wave inside
the barrier is more marked, and each time it reaches an interface, a small wave is released to
the outside. This way, apart from the widening of the pulse due to dispersion, the inner wave
gradually fades away.
Returning to the original example, we examine the trajectory of the peak of the wave packet
( g. 4.23). To determine this path, we havetwo possibilities to identify the maximum (see also
the discussion in section 1.5): a temporal and a spatial one. Here we consider the temporal
maximum, i. e. the moment in time for a given position in space when the observed eld reaches
its maximum amplitude. We recognise the linear motion of the incident and transmitted wave,
as well as the not surprising fact that a de nition of the peak's trajectory is not meaningful
in the interference zone in front of the barrier. Interestingly enough, the trajectory of the
transmitted wave is more strongly inclined towards the axis than the incident wave packet,
meaning that the packet to the right of the barrier runs faster. Again, this is largely due to
the spectral components that can propagate over the barrier, whereas the evanescent parts are
entirely attenuated by the thick obstacle. Hence the low-frequency components are ltered
out, and the wave packet emanating from the far side of the barrier now has a spectrum that
is narrower, but centred about a higher frequency than before. Here and in the subsequent
examples, the actual centre frequencies for both the incoming and transmitted wave packets
are computed from the trajectories and given in the captions of the respective gures. In the
current example, this centre frequency is even above the cuto frequency of the barrier. The
propagation of the high-frequency part is also responsible for the s-like shape of the trajectory
within the barrier and the o set between the extrapolated trajectories of the incident and
transmitted waves.
It is not di cult to calculate the actual centre frequency from the trajectories. Since the
wave packets are smooth and well-behaved, we may safely take the view of the phase time
supporters and assume that the wave proceeds at the group velocity, which wehavefound to
be
r
2~!
vg = :
m
(4.60)
It is easy to t a straight line into the space-time points of the trajectory outside the barrier
and away from the interference region, and from the normalised variables, we obtain directly
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