Today I continued the discussion of the scattering theory, deriving the Landauer formula, discussing the quantized conductance and the properties of scattering matrices. I was a bit slow, as I would have liked to finish the chapter on scattering, but I failed to show the derivation of the current from the current operator description, or the description of resonant tunneling.
It was first difficult to figure out clicker questions, but then I managed to find a few, with which I was quite satisfied. The first one was to figure out the average rate of electrons passing through a single-mode ballistic channel biased with voltage V. It was actually an exercise on dimensional analysis, as only one of the proposed alternatives had a dimension of rate. Anyway, I would like to teach this point in all courses: dimensional analysis in the sense of figuring out the correct dimensions (say, in 1/s or m etc.) is a very helpful tool in checking a result, or trying to find relevant quantities describing a certain system. Let’s say we want to study the dynamics of electrons in some system. We are hence interested in processes that change the number of electrons in a given region from time t to time t+dt, i.e., we should write a differential equation for this number. This means that the right hand side of this equation should have only quantities that have a dimension of rate. For a given model that we make on a given system, there are typically only a handful of such quantities, and only a few that are independent of each other. More often or not this number of quantities is one or two. In that case you can characterize the full dynamics by seeing what happens if you change the values of these quantities. This is where you get with a simple dimensional analysis!