Wave Propagation in Linear Media | re-examined

2 = !p
c
3 = !p
c
!p
c
Z 1
,1
Z 1
Z 1
,1
,1
A( )
A( )
p 2 , 1+ 2
4 One-dimensional quantum tunnelling
e
N( )
j(X,D)
p
2,1,jT 2
d +
p
2 2 , 1 ,
N( )
A( ) 2 p 2 , 1
N( )
e ,j(X,D)
p
2,1,jT 2
d ;
(4.42)
2
j(X,D) ,jT
e d ; (4.43)
where we set N( )=2 p 2 ,1 cos D p 2 , 1 , j(2 2 , 1) sin D p 2 , 1.
4.5 Tunnelling time de nitions for a square barrier
Before looking at the time-dependent evolution of wave packets impinging on the square
barrier of the previous section, we brie y discuss some well-known approaches to compute
the tunnelling time through such anobstacle. They all apply to monochromatic waves, but
have also been used to describe the behaviour of small-band pulses, in particular Gaussian
wave packets.
For a start, we consider the time it takes for a free particle to travel a distance d. This can
easily be found from the dwell time de nition (2.17), which with the monochromatic wave
= C1ej( x,!t) and the propagation constant = p 2m~!=~ yields
f = d
2
r
2m
: (4.44)
~!
We could obtain exactly the same result with the application of the group velocity de nition
and the `classical' argument that the propagation time is the distance d divided by the group
velocity 1=vg = p m=(2~!). With the normalised variables (4.12), this time becomes
f = d
p : (4.45)
2c
In the sequel, we shall use this undisturbed propagation delay of a free electron as a reference
for other concepts.
With a consideration similar to the aforementioned, we can derive a simple rst-cut approximation
to the barrier traversal time. Supposing that the particle moves freely inside a
potential barrier with the residual energy E , V0, we replace ! in the above expression by
! , !p and get the so-called semi-classical time [83, 86],
d
s =
2c p : (4.46)
, 1
Note that for waves with energy below the top of the barrier, this time is imaginary, a fact
that has widely been critisised. Referred to the free particle, the semi-classical time reads
s
f
=
r
96
: (4.47)
, 1