Wave Propagation in Linear Media | re-examined

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150
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T
4 One-dimensional quantum tunnelling
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Figure 4.34: Trajectory of the temporal peak of a wave packet with the parameters k = 20, n =4,
and = 0:2 impinging on a thick barrier (D = 10). The e ective centre frequencies are ci 0:2004
and ct 0:2028.
From the wave form de nitions in section 4.2, we nd the initial position of the peak,
X0 = k
p : (4.62)
The time the wave packet needs to travel this distance is readily obtained from the propagation
velocity of the free electron (4.45)
Tinc = X0 k
p =
2 2
: (4.63)
Since we want to explore the dependence of the tunnelling time on the carrier frequency, we
must ensure that the spectrum has equal width for all values of . The dominating factor in
the wave number spectrum (4.28) is e ,( k=n ( =p ,1)) 2
, whichs yields the relation
k
p = ; (4.64)
n
with some arbitrary, non-negative constant . The width of the initial wave packet, n, is held
constant, is our independent variable, so we must set k = n p = in the spectrum and
get the nal expression for the time of free motion,
Tinc = n
p : (4.65)
2
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