Wave Propagation in Linear Media | re-examined

4.5 Tunnelling time de nitions for a square barrier
The de nition of the dwell time has already been given in (2.17). For a rectangular barrier,
this time is found to be [83, 88]
D = mk
~
2 d( 2 , k2 )+k0 2 sinh 2 d
4k2 2 + k0 4 sinh 2 ; (4.48)
d
where in our notation k 2 =2m!=~, k0 2 =2m!p=~, and 2 = k0 2 , k 2 . Using scaled variables
and D = d!p=c, the expression can be written as
D
f
=2
1,2 + sinh 2Dp 1,
2D p 1,
4 (1 , ) + sinh 2 D p 1 ,
: (4.49)
Buttiker [88] derived a traversal time based on the idea of a quantum clock. This traversal
time is determined by the precession of the spin a particle experiences in a small magnetic
eld con ned to the barrier. This Buttiker time (or Buttiker-Landauer time, as it was called
in the survey of Hauge and St vneng [85]), consists of two components,
with the dwell time d given by (4.48) and the Larmor time
z =
mk0 2
~ 2
In normalised notation, this reads
z
f
=
r
1 ,
B = p d 2 + z 2 ; (4.50)
( 2 , k2 ) sinh 2 d + 1
2 dk0 2 sinh 2 d
4k2 2 + k0 4 sinh 2 : (4.51)
d
(1 , 2 ) sinh2 D p 1,
D p 1,
+ 1
2 sinh 2Dp 1 ,
4 (1 , ) + sinh 2 D p 1 ,
: (4.52)
The last one of the tunnelling time de nitions considered here is the phase time, which
essentially describes the movement of the peak of a wave packet. For our square barrier, it is
given in the literature [83] as
p = m
~k
which can nally be rewritten as
p
f
=2
2 dk2 ( 2 , k2 )+k0 2 sinh 2 d
4k2 2 + k0 4 sinh 2 ; (4.53)
d
(1 , 2 )+ sinh 2Dp 1,
2D p 1,
4 (1 , ) + sinh 2 D p 1 ,
: (4.54)
Prior to comparing these four traversal times for two selected cases, we add a fth based
on the dwell time de nition. We recall that the dwell time was de ned as the ratio of the
probability density in the barrier and the incident ux, integrated over the barrier region.
The incident ux, however, contains also a portion that will be re ected from the barrier, and
it is by no means evident why this re ected wave should be taken into account when a time
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