Problem 1
Many observable properties of a metal can be calculated from the $E$ vs. $k$ dispersion relation.

(a) How could the electrical conductivity be calculated from the dispersion relation?

(b) How could the dielectric function be calculated from the dispersion relation?

(c) How could the electron dispersion relation of a metal be measured?

Problem 2
Would you expect to observe plasmons, polarons, polaritons, and excitons in a metal or an insulator?

Could plasmons, polarons, polaritons, and excitons be observed by Raman spectroscopy? EELS? Ellipsometry?

Problem 3
The dielectric constant of materials used in a supercapacitor can be high ($\epsilon_r$~1000). Explain why the dielectric contant is temperature dependent. The index of refraction $n=\mathcal{Re}\left[\sqrt{\epsilon_r} \right]$ is not so large for these materials ($n$~2). Why is this so?

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.