Wave Propagation in Linear Media | re-examined

τ
8
7
6
5
4
3
2
1
4 One-dimensional quantum tunnelling
0 0.5 1 1.5 2 Ω
Figure 4.19: Tunnelling times referred to the propagation time of a free particle, f , for a thick barrier
(D = 10). The curves denote the phase time p (solid line), dwell time d (dotted line), Buttiker-
Landauer time B (dashed line), the absolute value of the semi-classical time j sj (dot-dashed line),
and the e ective dwell time e (thick solid line).
measure for the traversal through the barrier is sought. Therefore we try a di erent approach
and use only the wave function inside the barrier for the computation of the ux.
In principle, we follow the same path as in the calculation of the electromagnetic energy
velocity in section 3.5 . The local velocity of the wave then, is the ratio between the ux
J = j~ ,
r , r : (4.55)
2m
and the probability density averaged over a period in time,
v = J
(4.56)
Inside the barrier, we must use 2 from (4.36). After a lengthy calculation, we end up with
the reciprocal of the local velocity, a local transmission time
1
v =
s 2m
~(!p , !)
2 + 2
8
e ,2 (x,d) + e 2 (x,d) +2
2 , 2
2 + 2
; (4.57)
with = 1 = p p 2m!=~ as the propagation constant outside the barrier and = ,j 2 =
2m(!p , !)=~ as the one within, respectively. To obtain the e ective overall traversal time,
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