Scattering and interference

From the title one might get the idea that the lecture today was pretty messed up, with people interfering with my lecturing a lot. Unfortunately, it was pretty smooth with only a couple of (admittedly good) questions. A ballistic lecture with no backscattering is never a good idea, because no information can be inferred from the lecture that way.

Jokes apart, I seem to be always late from my schedule, as I had hoped to get a bit further in the interference chapter. I did discuss at length the treatment of many-probe systems in scattering theory, showed examples of the voltage probe and the comparison between two- and four-probe resistances. Then I discussed the resonant tunneling effect, which is probably somewhat familiar from quantum mechanics I. In fact, I think that the scattering formulation is the best way to describe this effect: it is generic (describes both the quantum dot case and the case of Fabry-Perot cavities in optics), the only differential equation one needs to solve is the one related with the dynamic phase, and it shows the difference in summing the probability amplitudes or probabilities for the particle paths. Moreover, it connects waves with quantized levels.

In the scattering formulation, this along with the four-probe conductance formula allows showing how to get localization-type of scaling of conductance in a disordered wire. Note that this was debated a bit in the 1970’s-1980, as the proper result depends on which quantity is averaged.

Anyway, I only reached mid-way of weak localization, so next time I will have to deal with the negative magnetoresistance related to it, and discuss a bit universal conductance fluctuations. Persistent currents I will probably leave for the self-study, as I want to be able to discuss superconductivity also during the next lecture.

After the lecture I got a question about notation in the book. It seems that in chapter 3 I am using two notations for the number of channels in a lead: M and N. M is the number of modes, N denotes the size of the reflection matrix, but these two are of course the same.