Wave Propagation in Linear Media | re-examined

4.3 Examples of scattering processes
and determine the fraction Pe=Pt for the various inital waveforms. To plot this ratio, however,
it is more suitable to regard the complementary quantity, i. e. the part of the probability that
still originates from the pass band,
P =1, Pe
Pt
: (4.32)
Evaluations of this relation are depicted in g. 4.4 in dependence on the centre frequency and
for di erent lengths of the wave packet.
The plots reveal that even for waves comprising a large number of carrier wavelengths, the
pass band still contributes a signi cant part of the total probability ifwe start with a square
wave. The situation is much better for triangular waves that even surpass narrow Gaussian
wave packets for frequencies slighty below cuto .
4.3 Examples of scattering processes
In this section we nally examine several examples of scattering events of electrons impinging
on a step potential barrier. The partial waves outside and inside the barrier are de ned by
the Fourier integrals (4.14) { (4.16). The appropriate wave number spectra of the initial wave
forms are taken from section 4.2, and the integrals are evaluated numerically. The plots always
show the probability density function P = j j depending on the normalised variables of
space and time, X and T .
Let us begin with an initially rectangular wave like in g. 4.1. As we have already seen, a large
portion of its spectral components lies in the pass band, which makes such a wave a rather
unfortunate choice for an investigation of the tunnel e ect. It is, however, useful to study the
e ects of partial transmission of the high-frequency components. But before we do so, let us
brie y consider the wave packet as such. Owing to its wide spectrum, we must expect that it
will soon lose the original shape and take on a smoother form. This suspicion can easily be
con rmed by evaluating the incident wave(4.14) not only to the left of the barrier, but also
in the right half space, where it corresponds to a particle moving freely without obstacle. The
re ected and transmitted parts of the wave are therefore not of concern in this case. Fig. 4.5
shows indeed that dispersion quickly changes the shape of the wave packet from the initial
rectangle to a broad pulse with a marked peak and a number of small sidelobes.
Remark (Plot ranges) In the sequel, the three-dimensional plots of the probability
density outside the barrier are always accompanied by a contour plot of the same function.
To allow the perception of as many details as possible, the plots often deliberately do not
show the full range of function values. This is particularly important for the contour plots
to avoid an exceedingly large number of contour lines. The bright white holes in these
plots thus indicate the areas that have been clipped.
Now let this particle collide with the step potential. The plot of the probability density
outside the barrier ( g. 4.6) shows the incident wave packet moving towards the barrier while
already losing its shape. The actual scattering event is characterised by strong interference
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