Wave Propagation in Linear Media | re-examined

17.5
15
12.5
10
7.5
5
2.5
τ
0.5 1 1.5 2 2.5 3 Ω
35
30
25
20
15
10
5
τ
4 One-dimensional quantum tunnelling
0.5 1 1.5 2 2.5 3 Ω
Figure 4.37: Numerically determined tunnelling times compared with the phase time theory for a thick
barrier (left graph: D = 10, right graph: D = 20) and small-band electrons ( = 100, n = 4).
The frequency shift may be small, but it always occurs. The relative variation of the group
velocity, on the other hand, is much larger for very small energies because the group velocity
approaches zero in the DC-limit, and so the acceleration e ect the wave packet experiences
is larger, too.
Remark (Trajectories inside the barrier) The above explanation becomes even
more elucidating if we consider how the trajectory inside the barrier emerges. We already
pointed out (see section 4.3) that it is the result of the successive arrival of the wave
packet's components, all travelling at distinct velocities. The rst to arrive are the highenergy
parts that also have a higher tunnelling probability. Hence comes the inclination
of the trajectory towards the axis. The steady-state tunnelling probability corresponds to
the stationary penetration depth, which naturally is very small for low energies, so that
its relative increase due to a xed frequency shift is much larger for low than for high
energies.
The considerations given above also anticipate what will happen if we choose a broader
spectrum for the incident wavepacket. Fig. 4.36 shows the results for a thick barrier, where
the e ect is even more pronounced. The left graph was drawn with the same parameters we
chose for calculating the trajectories of the low-energy electrons earlier in this section. We
recognise that for . 0:5, the numerically determined tunnelling time becomes negative,
which is in line with the observations we made when computing the trajectories. Likewise,
there is no resemblance at all between the numerical results and the phase time theory |
on the contrary, they are rather close to the simple semi-classical time in that they do not
exhibit any resonance peaks above cuto and have a maximum exactly at = 1, in contrast
to all other de nitions.
The situation changes if the bandwidth of the impinging electron is reduced like in the right
picture in g. 4.36, where the parameters of the trajectory in g. 4.34 were used. Finally, in
g. 4.37 we selected again an incident wavepacket with a `properly' narrow bandwidth and
obtain the expected excellent agreement between numerics and the phase time prediction.
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