Problem 1
(a) What are polaritons?

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

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

(d) How could you measure polaritons?

(e) Do you observe polaritons in metals, semiconductors, or insulators? Why?

Problem 2
When using the Boltzmann equation to calculate currents, a function $f$ is defined which gives the probability that state $\vec{k}$ is occupied at position $\vec{r}$ and time $t$.

(a) Give an expression for the electrical current density in terms of $f$ and the density of states $D(\vec{k})$.

(b) Write down the Boltzmann equation that must be solved to find $f$. (Hint: Take the total derivative of $f$.)

(c) What is the meaning of the collision term in the Boltzmann equation?

Problem 3
Ferromagnetism can be described by Landau's theory of second order phase transitions.

(a) What is the order parameter for ferromagnetism? Sketch the temperature dependence of the order parameter.

(b) How could you observe this phase transition in an experiment?

(c) How could you calculate the magnetic susceptibility from Landau theory? Sketch the temperature dependence of the magnetic susceptibility.

Problem 4
Describe a Mott transition and a Peierls transition. What properties would a material need to have to exhibit a Mott transition or a Peierls transition?