S. Scandolo, ICTP

Lesson 13 (Crystals: the quasi-free electron model)

The lesson starts with continuing the discussion of the free electron approximation from last class. The Fermi energy for 3 dimensions is derived.

As an interlude, the professor talks about the natural units and scales of the Hamiltonian. The comparison between the kinetic and potential energy terms gives a characteristic length, the Bohr radius. Also, a characteristic energy, Hartree energy (Ha), can be associated with this length. This is related to another unit of energy, the Rydberg constant as 2Ry = 1Ha.

One step beyond the free electron model of crystals, there is the quasi-free electron model. In this model the potential is weak and the solutions of the Hamiltonian can be treated as perturbed solutions of the free electron model. Before solving the model, the professor gives a brief recap of perturbation theory and defines the normalization of the free wave solutions to be 1 in the unit cell. The change of energy levels for non-degenerate states in the first order are given by calculating the diagonal components of the potential in the free wave basis. If there are degenerate energy values, one needs to use eigenfunctions which diagonalize the potential in their subspace.