Wave Propagation in Linear Media | re-examined

4.4 The square barrier
they have vanished, and only the over-the-barrier contributions remain. Even the narrow
Gaussian pulse breaks up into distinct oscillations. Only the originally wide Gaussian wave
packet retains its shape, of course with a dramatically reduced amplitude. So it is just this
last case where we can assume the spectrum to have been narrow enough and su ciently
below the top of the barrier. Note also that the amplitude of the narrow pulse is about
two orders of magnitude larger than that of the wide pulse, which undeniably identi es the
high-frequency components.
The last examples explore a longer wave packet (k = 20) at the same positions. At the
interface ( g. 4.16), the situation has not changed very much. There are still oscillations,
although with a lower amplitude, in the rectangular pulse, and its shape can now be identi ed
much easier than before. The skewness of the Gaussian pulses is much smaller now, according
to the narrower spectrum. Within the barrier ( g. 4.17), it is interesting to notice that the
rst peak of the rectangular wave packet has about the same height as for the short electron
in g. 4.15. This demonstrates once more that the number of carrier wavelengths the particle
embraces has no great in uence on the fraction of pass-band spectral components, as can
also be seen from g. 4.4 . The oscillations that remain from the triangular wave packet,
however, have become smaller as compared with the short particle before. But the most
striking di erence is the behaviour of the narrow Gaussian pulse, whose spectrum this time
is also entirely below the barrier level. Consequently, the relation between the amplitudes
inside the barrier is roughly the same as at the interface.
4.4 The square barrier
The investigation of the penetration of an electron into a step potential barrier provided
already some insight into the behaviour of evanescent waves in quantum mechanics. It did
not, however, give an answer to the question of the tunnelling time of a particle, because an
in nitely extended step barrier is of course always opaque and permits no tunnelling through.
Neither can the results of the step potential be easily extrapolated to a square barrier due
to the presence of a second interface that will inevitably cause re ections and interference.
There is thus no choice other than to examine also the case of a square potential barrier
V (x) =
8
><
>:
0 if xd
: (4.34)
Since there are now three distinct regions, we have three waves to determine. To the left of
the barrier, there is the incident and a re ected part,
1 = C 1 e ,j!t e j 1x + C2 e ,j!t e ,j 1x : (4.35)
Inside the barrier, also a right- and a left-going solution will exist,
2 = C 3 e ,j!t e j 2x + C4 e ,j!t e ,j 2x ; (4.36)
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