Wave Propagation in Linear Media | re-examined

tun = !p
c
+ !p
c
+ !p
c
Z ,1
,1
Z 1
,1
Z 1
1
A( )
A( )
A( )
4 One-dimensional quantum tunnelling
2
, p e
2 , 1 ,jX
p
2,1,jT 2
d +
2
+ j p 1 ,
2 e,X
p 1, 2 ,jT 2
d +
2
+ p e
2 , 1 jX
p
2,1,jT 2
d :
(4.16)
These equations describe the complete scattering process independently of the initial wave.
Remark (Energy and momentum space) From electromagnetic waves we are used to
decomposing wave packets either into a frequency or wave number spectrum. Throughout
the remainder of this chapter, we shall adhere to these terms also for quantum mechanical
wave packets. It should be noted, however, that quantum physicists more commonly talk
about energy and momentum spectra. This seems a bit confusing, but in fact is equivalent
to our terminology. The energy space corresponds to the frequency spectrum because of
the relation E = ~!. The momentum in quantum mechanics is de ned as p = ~ and
therefore corresponds to our usual wave number space. So, apart from the factor ~, both
views are identical.
4.2 Initial wave forms
There are two mutually exclusive choices for the initial conditions. First, we can restrict the
location of the wave packet to the space outside the barrier. In this case, the spectrum will
inevitably contain components with an energy above the barrier. Conversely, we can as well
limit the bandwidth of the wave packet to energies below the barrier, but then the wave will
not be bounded in space and therefore extend into the barrier from the beginning. We deem
this more undesirable and thus choose the rst possibility.
We now calculate the spectra of three di erent wave forms to be used as initial conditions
in a scattering process. The position of the wave packets is a snapshot of the very moment
when they reach the barrier.
The rst and simplest one is a rectangular pulse. It is given by
(
0 if x