Problem 1 (a) Explain why the Pauli exclusion principle is only valid for systems of non-interacting particles. (b) Exchange is a quantum many-body effect. Explain why the exchange energy is zero for systems of non-interacting particles. (c) How does exchange lead to ferromagnetism? Problem 2 The discussion of the quantum Hall effect and the discussion of superconductivity both started with the quantum description for an electron in a magnetic field. The Schrödinger equation for this case is, (a) How is it possible that we started with the same Schrödinger equation but ended up with two completely different phenomena (the quantum Hall effect and superconductivity)? (b) Describe the phase transistion that superconductors undergo. Include the free energy and the entropy in your discussion. (c) Describe the phase transition that ferroelectrics undergo. Include the free energy and the entropy in your discussion. Problem 3 Consider two insulating materials: one shows polaritons and the other does not. (a) Sketch the dielectric function of both materials. Explain why you have drawn the dielectric functions the way you have. For the material that exhibits polaritons, indicate the frequencies for which polaritons would be observed. (b) What would Raman spectroscopy tell you about the phonons of these materials? (c) What would an inverse photoemission spectroscopy tell you about these materials? Problem 4 (a) What does this tell you about the properties of Bi? (b) How would you calculate the Fermi surface? (c) How can you tell if the charge carriers are electrons or holes? (d) How can you measure the Fermi surface experimentally?
Bismuth is a semimetal with a very small Fermi surface.