Wave Propagation in Linear Media | re-examined

-60 -50 -40 -30 -20 -10
3
2
1
-1
Ψ0
X
4 One-dimensional quantum tunnelling
A
0.14
0.12
0.1
0.08
0.06
0.04
0.02
-2 -1 1 2 ξ
-2 -1 1 2 ξ
Figure 4.2: Triangular pulse with k = 6 and = 0:8. The left graph shows the real part of the
wave function (thin line) and the corresponding probability density (thick line). The right graph gives
the absolute value jA( ) l ,1=2 j of the wave number spectrum (4.25), peaked about = 1=2 . The
evanescent region is the interval [,1; 1].
Like before, the factor p 3=l is due to the normalisation R 1
easily found to be
or, in scaled variables,
A tria ( )
p l
p
3
=
4 3k2 Atria( )= p p
3
l
4
1
1 , 1
p
,1 0 0
dx =1. The spectrum is
sin( 0 , ) l
4
( 0 , ) l
! 2
e
4
,j( 0, ) l 2 (4.24)
,j k 1, p
1
2e 2
The wave packet and its spectrum are shown in g. 4.2 .
,j2 k 1, p
1
, e
, 1 : (4.25)
The last example is also the most popular in the literature: the Gaussian wave packet. If we
start with a normal distribution N( ; 2 ) for the probability density and choose = ,l=2,
= l=(2n), we get the wave function
0 (x) =
s
2n
l p 2 e,n2 ( x l +1 2) 2
e j 0x
: (4.26)
As the envelope function is not truncated at x =0,care must be taken to set n su ciently
large so that the wave function almost vanishes inside the barrier. The spectrum of this wave
packet is simply
Agauss ( )= p 1
lp
p e
2 n 2 , ( 0
78
, )l
2n
2
,j( 0, ) l 2 (4.27)