Wave Propagation in Linear Media | re-examined

4.6 Examples of tunnelling events
The two graphs also reveal that the deviation between the two curves deteriorates as the
barrier thickness is increased.
The conclusions to be drawn from the above are:
The barrier acts like a high-pass lter favouring the high-energy components of the
incident wave packet. Thus the transmitted portion of the wave experiences a frequency
skewing and a shift of its e ective centre frequency to higher values. This pulse reshaping
e ect occurs always, no matter how narrow-band the wave packet originally was, and
it is more marked for low-energy waves.
The phase time theory is in good agreement with numerically found tunnelling times,
provided that the wave packet exhibits a dominating peak that can be traced and
taken as a measure for the propagation. Furthermore, the wave packet must also be
very narrow-band so that the group velocity approximation for the propagation velocity
holds. The spectrum need not be con ned to the evanescent region, and it may also
extend into the pass band.
The phase time approximation becomes worse for very low energies of the incident
electron due to the relatively severer e ect of pulse reshaping in this region.
If the initial wave packet is not small-band, the phase time gives no appropriate description
of the tunnelling time except for very large energies in the pass band. If the
tunnelling time is still based on the evolution of the peak of the wave packet, even
negative delay times may be obtained. It is questionable whether the de nition of the
tunnelling time is still sensible then.
As for the faster-than-light debate, we stress that the foregoing examination was based
on the non-relativistic Schrodinger equation. In the entire chapter, the velocity of light
appeared not a single time. So in order to judge how fast the particles or waves actually
travel, we would need to take the theory of relativity into account. Note also that in
contrast to the electromagnetic case, the quantum mechanical wave function has no nite
wave front velocity. The reason for this di erence is the quadratic dispersion relation
of quantum waves, whereas those of wave guides and lossless plasmas are hyperbolic.
This exposes the limitation of the analogy between undersized wave guides and square
barriers: it is valid as long as the respective di erential equations are regarded. As one
looks into their solutions, however, it breaks down.
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