Problem 1
(a) What are plasmons?

(b) How would you calculate the dispersion relation ($\omega$ vs. $k$) for plasmons?

(c) Make a plot of the plasmon dispersion relation. Indicate roughly the values of $\omega$ (in Hz) and $k$ in (1/m) that you expect the plasmons will have.

(d) How could you measure plasmons?

(e) Do you observe plasmons in metals or insulators? Why?

Problem 2
A two-dimensional metal has a rectangular Bravais lattice with primitive lattice vectors:

The metal has three valence electrons. Draw the Fermi surface indicating if the states are electron-like, hole-like, or open orbits.

Problem 3
What are Shubnikov-de Haas oscillations? Why do they occur? What other quantities exhibit oscillations as the magnetic field is varied? How can the oscillations be used to experimentally determine the fermi surface of a metal?

Problem 4
A metal-insulator transition can be induced by electron-electron interactions.

(a) Explain why electron-electron interactions are difficult to describe.

(b) A simple model for electron-electron interactions is electron screening. Explain what screening is and how it can be used to explain a Mott transition.

(c) Single electron effects are another simple way to include electron-electron interactions. Explain the single-electron charging effect.