Wave Propagation in Linear Media | re-examined

P
0
2
1.5
1
0.5
0
0
1
50
2
X
T
100
3
150
4
4 One-dimensional quantum tunnelling
Figure 4.11: Evolution of the probability density P = j j of a longer triangular wave packet within
the barrier. The parameters are k = 10 and = 0:2.
realistic case of a longer electron with triangular shape. In addition, we choose a carrier wave
with lower energy. The contribution of the pass band to the total probability density inthe
initial wave form is then low enough that we may expect `proper' penetration of the wave
into the barrier. This reduces the disturbing e ects of the transmitted components, although
we will of course never really get rid of them. Since the wave spans more wavelengths of the
carrier, interference between incident and re ected part generates more peaks, as can be seen
from g. 4.10 .
Inside the barrier, the probability density is largely determined by evanescent components,
so that the overall shape of the function decays exponentially ( g. 4.11). However, a closer
inspection of the probability would again reveal high-frequency ripples covering the surface at
least for small values of T . Owing to the choice parameters, they are very small in comparison
with the evanescent part, but nonetheless become prominent for large values of X, aswe shall
see shortly.
Remark (Evolution of the peak) The contour plots of the scattering events give
a bright illustration of the di culties associated with the phase time concept. In this
approach, the trajectory of the peak of the wave packet determines the tunnelling time.
But as the plots show, it is impossible to identify the peak in front of the barrier because
of the interference of incident and re ected wave. Hence the trajectory can only be
extrapolated from the evolution of the wave packet far away from the barrier where
interference has not yet distorted its shape (see also the respective remark in section 2.2).
The last example shows a Gaussian wave packet with a narrow spectrum. The parameters
are the same as those used in g. 4.4 , so that the fraction of the pass-band components is
given by the left graph in g. 4.4 . The plots of the scattering process in front ofthebarrier
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