Wave Propagation in Linear Media | re-examined

4 One-dimensional quantum tunnelling
spectrum in wave number space, A( ), each of the monochromatic components is itself a
steady-state solution of Schrodinger's equation, and the superposition of all partial waves
then yields the desired evolution of a single wave. The general form of this solution is
(x; t) =
Z 1
C A( ) e
,1
j( x,!( )t) d ; (4.7)
where C and also have tobechosen according to the respective region. The incident wave
then reads
inc =
Z 1
,1
j x,j
~
A( ) e 2m 2t d : (4.8)
The formulation of the re ected part is not so straightforward and requires a moment's
consideration. The re ection coe cient contains the propagation constants of both the region
outside the tunnel and the tunnel itself. The link between the respective dispersion relations,
however, is provided by the frequency ! that alone remains invariant as the monochromatic
wavelets penetrate into the barrier, whereas the propagation constant changes. Therefore we
must not mix up 1 and 2. Instead, we retain 1 as and express 2 by means of (4.6)
as 2 2 = 2 , 2m!p=~. Consequently, the integration range falls into three distinct intervals
determined by
2 =
8
><
>:
,
q 2 , 2m!p
q 2m!p
~ if
~
q
2m!p
~
q
2m!p
~
q
2m!p
~
q 2m!p
~
: (4.9)
Remark (Transformation of ) The use of the negative rootinthetransformation
becomes obvious if we call to mind that the dispersion relations are even functions in and
separated in ordinate direction only by the cuto frequency !p. Hence the transformation
rule must be even, too, which gives 2 = sign p 2 , 2m!p=~ outside the evanescent
region.
With (4.9) and (4.4), we nally obtain the three parts of the re ected wave,
ref =
Z ,
+
,1
q 2m!p
~
Z q 2m!p
~
q 2m!p
, ~
Z 1
+ q 2m!p
~
A( ) +
A( )
,
q
2 2m!p
, ~ q
2 2m!p
, ~
q
2m!p
, j ~ , 2
q
2m!p
+ j ~ , 2
A( ) ,
q
2 2m!p
, ~
+
q
2 2m!p
, ~
74
,j x,j
~
e 2m 2t d +
,j x,j
~
e 2m 2t d +
,j x,j
~
e 2m 2t d :
(4.10)