Wave Propagation in Linear Media | re-examined

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4
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4 One-dimensional quantum tunnelling
0 100 200 300 400 T
Figure 4.16: Pulse shapes produced by several long wave packets at the interface of the barrier. The
parameter are k = 20 and = 0:2. The left graph shows the rectangular (solid line) and triangular
(dotted line) packet. In the right graph, the results of a narrow (n= 10, solid line) and a wide (n =4,
dotted line) Gaussian packet are given.
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Figure 4.17: Pulse shapes produced by several long wave packets at X = 10 inside the barrier. The
parameter are k =20and =0:2. The graphs show the rectangular (left), triangular (middle), and
Gaussian (right) packet. In the latter, the results for a narrow (n= 10, solid line) and wide (n =4,
dotted line) packet are displayed.
we nowchoose the parameters so as to obtain as narrow a spectrum as possible. To permit a
direct comparison of the spectra, we again take the parameter values of g. 4.4 together with
alow-frequency carrier ( = 0:2).
Fig. 4.14 shows the pulses of a short electron (k = 6) at the interface of the barrier. Not
surprisingly, the rectangular wave packet exhibits strong oscillations. As a direct consequence
of dispersion, the temporal frequency of these oscillations decreases rapidly in the course of
time. On the other hand the triangular pulse, though having also spectral components in the
pass band, shows no such oscillations. The graph also reveals that the Gaussian wave packets
do not produce Gaussian pulses, but wave forms with a remarkable skewness. This e ect is
more marked if the wave packet is narrow and therefore has a broad spectrum, as explained
above.
Inside the barrier ( g. 4.15), the wave forms undergo signi cant changes. For the rectangular
wave packet, the high-frequency components that can propagate into the barrier become
more pronounced. This also goes for the triangular packet, which had looked so smooth
at the entrance. There the evanescent parts had obviously dominated the shape, but now
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