Wave Propagation in Linear Media | re-examined

4.1 The potential step
Inside the barrier, we must again apply (4.9), although one might perhaps argue that in this
region the original 2 would have been more appropriate. This would in fact be true if the
spectrum of the initial wave, A( ), had been given in terms of the wave numbers inside the
tunnel. But as it is, the outset of our investigation is a wave strictly con ned to the outside of
the barrier, and so we cannot but describe it in that wave number space. Thus the transmitted
part of the wave becomes
tun =
Z ,
+
,1
q 2m!p
~
Z q 2m!p
~
q 2m!p
, ~
A( )
A( )
Z 1
+ q A( )
2m!p
~
,
+
2
q 2 , 2m!p
~
+ j
2
q 2m!p
~ , 2
2
q 2 , 2m!p
~
e ,jx
e jx
q
2, 2m!p
~ ,j ~
2m 2 t d +
e ,x
q 2m!p
~ , 2 ,j ~
2m 2t d +
q
2, 2m!p
~ ,j ~
2m 2t d :
(4.11)
As usual, we introduce normalised variables to ease the numerical evaluation,
r
~ !p
c =
2m ; T = !p t; X= !p
x; (4.12)
c
as well as a new integration variable
=
s ~
2m!p
= c
!p
: (4.13)
Note that in this context, c has nothing at all to do with the velocity of light. It is nothing
butavariable that incidentally has the physical dimension of a velocity. The individual wave
functions can be rewritten as
inc = !p
c
ref = !p
c
+ !p
c
+ !p
c
Z 1
A( ) e
,1
jX ,j 2T d ; (4.14)
Z ,1
,1
Z 1
,1
Z 1
1
A( ) + p 2 , 1
, p ,jX ,jT 2
e d +
2 , 1
A( ) , jp1 , 2
+ j p ,jT 2
e,jX d +
1 , 2
A( ) , p 2 , 1
+ p ,jX ,jT 2
e d ;
2 , 1
75
(4.15)