Today's Class.

I hope you have all the sign issues sorted out and are getting some interesting results for \(\chi (T)\), in the normal state, and \( m(T)\) below the temperature at which spontaneous magnetization emerges, \(T_c\). Your results should take the form of graphs, with very well-written figure captions. Additionally, there should a discussion, but make the figures, the graphs, central to the readers experience.  I can't make it to class today, so you can turn those in to my mailbox anytime tomorrow if you like.

 Regarding class today,  would you prefer to discuss susceptibility and magetization or, perhaps better, work on the matrix we were endeavoring to elucidate last class. The matrix is very important. It provides the key to exploring a lot of physics including the physics of localization, and the physics of electron-electron interaction in a more general and powerful approach than the one we used in our model of magnetism. The matrix provides a context in which to question our assumption that the states of a crystal are always Bloch states. (That is, states with q-dependent Bloch phase factors, states that extend through the entire crystal and are 100% itinerant (no home).) There are numerous Nobel prizes in that matrix and other that we can evolve from it.

Perhaps once you get that matrix you can use it to look closely and carefully at the time dependent evolution of an electron that is initially, at t=0, in a localized state. How does \(\Psi(t)\) evolve as a function of time (for the first two "clicks" or so)? How does that depend on Bandwidth? How is bandwidth related to the elements of your Matrix?