Wave Propagation in Linear Media | re-examined

4.3 Examples of scattering processes
P
30
2
1.55
1
0.5 .5
0
0
20
10
X
10
20
T
0
30
Figure 4.8: Evolution of the probability density P = j j of a triangular wave packet within the
barrier. The parameters are k =3and =0:8.
40
barrier energy T =1b= X=1b=
0:3eV=4:8 10 ,20 J t =2:194 fs x =1:42 nm = 14:2A
0:1eV=1:6 10 ,20 J t =6:582 fs x =2:46 nm = 24:6A
Table 4.1: Actual dimensions for typical barriers.
We now move on to triangular electrons, the spectrum of which is far more well-behaved
than that of a rectangular particle. For the rst example we use the same parameters as
in the previous case. The behaviour outside the barrier ( g. 4.9) is essentially the same as
before, but the wave forms are much smoother now. The sharp peak of the initial wave
packet naturally disappears after a short time, and the re ected wave already resembles a
Gaussian distribution. The same observation can be made for the part that penetrates the
barrier ( g. 4.8). Since the parameters were chosen such that a fairly large contribution still
comes from the pass band, we recognise a soft crest moving ahead. Actually, the smoothlooking
wave function is superposed with small high-frequent ripples that, like in the previous
example, cannot be resolved in this plot.
The wave packets considered so far were extremely short (i. e. comprising very few wavelengths
of the carrier), which consequently resulted in a broad spectrum. Let us now turn to the more
85