Wave Propagation in Linear Media | re-examined

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4 One-dimensional quantum tunnelling
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Figure 4.23: Trajectory of the temporal peak of a wave packet with the parameters k =6,n= 4, and
=0:8interacting with a barrier of thickness D = 10. The e ective centre frequency of the incident
wave is ci 0:8181, for the transmitted wave itis ct 1:11.
a relation to the carrier frequency !c=!p = c,
T
X
c t 1
= =
vg t 2 p : (4.61)
c
Remark (Calculation of the centre frequency) For the incident wave, the e ective
centre frequency c should of course be equal to the carrier frequency selected for the
initial wave. The value obtained from the trajectory, however, is slightly di erent, and
experiments show that for a trajectory consisting of spatial maxima, the correspondence
would have been better. This is not surprising as the initial wave packet was de ned in
space rather than in time. Though being fairly close together, the spatial and temporal
maxima are still slightly di erent.
It should also be noted that the centre frequency calculated from the trajectory is not the
peak frequency of the spectrum, as one could suspect. According to (4.43), the spectrum
of the initial pulse is ltered through the transmission coe cient, which introduces a
skewness to the originally symmetric distribution. In the particular case of = 0:8 , the
spectrum is even oscillatory, because of the resonance peaks of the transmission coe cient
above cuto (see also g. 4.18).
Figs. 4.24 and 4.25 show the situation for the unaltered initial wave packet, but with a decreasing
width of the barrier. The e ects the obstacle exerts on the wave remain in principle
the same as before, with one great exception: there is no residual wave inside the barrier after
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