Wave Propagation in Linear Media | re-examined

0.1
0.001
0.00001
-7
1. 10
-9
1. 10
0.1
0.001
0.00001
-7
1. 10
-9
1. 10
ΔΡ
ΔΡ
0.2 0.4 0.6 0.8 1 Ω
0.2 0.4 0.6 0.8 1 Ω
0.1
0.001
0.00001
-7
1. 10
-9
1. 10
0.1
0.001
0.00001
-7
1. 10
-9
1. 10
ΔΡ
ΔΡ
4 One-dimensional quantum tunnelling
0.2 0.4 0.6 0.8 1 Ω
0.2 0.4 0.6 0.8 1 Ω
Figure 4.4: Fraction of the probability contribution from the pass-band spectral components (4.32)
for rectangular (solid line), triangular (dotted line), and Gaussian (dashed line) initial wave forms and
short waves. The parameters are k = 6 (upper left diagram), k = 20 (upper right diagram), k = 100
(lower left diagram), and k = 1000 (lower right diagram), respectively. The Gaussian wave packet was
computed for a wide (n = 4, right dashed line in each picture) and a narrow (n= 10, left dashed line)
example.
For the Fourier transform pair (4.7) and (4.18), Parseval's theorem reads
Z 1
,1
j (X)j 2 dX =2
Z 1
,1
jA( )j 2 d : (4.30)
Owing to normalisation, these integrals should give unity. The reason why (4.29) still
di ers from this value is simply the use of the scaled variables, where the factor p l could
not be transformed. If we had calculated the probability from the original spectra A( ),
we would have obtained unity as expected. For the ratio of evanescent contribution and
total probability we are interested in, the constant factor is of course irrelevant and cancels
out.
We now calculate the probability contribution from the spectral components in the evanescent
region,
Pe =
Z 1
,1
jA( )j 2 d ; (4.31)
80