# Homework #1

*due 5pm Friday of first week of class*

## Frequency dependent polarizability (10 pts)

The frequency dependent polarizability of an isolated hydrogen atom is a useful phenomenological model for understanding many “below bandgap” effects.

**a)** Use a simple “electron on a spring” model and a sinusoidal driving force *eE*_0*sin(ωt), to show

where the restoring force on the the electron is described by a spring constant *K* = *m*ω_0^2.

**b)** Find the dc polarizability of a hydrogen atom (I'm looking for a number here) using the result

- For 5 bonus points, complete A&M Ch27 Q3 to prove Eq. 2.
- Hints for A&M Ch27 Q3: To first order, the expectation value for kinetic energy (p^2/2m) is unchanged by the electric field. Therefore, focus on finding the lowest potential energy for the trial wavefunction.

**c)** What is ω_0 and how does it compare to the frequency of visible light?

**d)** Use a computer to plot α vs log(ω) for ω = 0.1ω_0 to 10ω_0. Choose your y axis so that the difference between α(0.1ω_0) and α(10ω_0) is clear.

## Solid hydrogen in a cubic lattice (10 pts)

*(this problem is similar to the derivation of the Clausius-Mossotti relation in A&M)*

The polarizability of an isolated atom differs from the polarizability of an atom that is embedded in a lattice. Consider a cubic array of hydrogen atoms with lattice spacing *a*. The hydrogen crystal fills a rectangular box which has two longs sides and one short side (a slab geometry). An external electric field E_ext is applied perpendicular to the large faces of the slab. Each H atom feels E_ext plus the E-field generated by other polarized atoms in the lattice.

**a)** Choose an arbitrary hydrogen atom near the center of the crystal. Show that the local electric field at the coordinates of the hydrogen atom is E_local = E_ext - 2P/3ε_0, where P is the bulk polarization of the sample. You will have to sum the fields from individual dipole moments near the hydrogen atom (see Ashcroft and Mermin p539-541), then add the average field from all the remaining dipoles in the slab.

**b)** Using part a), show that the dielectric constant of this material (ε = E_ext/E_avg for a slab geometry) diverges when the spacing between H atoms is 2.7 time the Bohr radius.

## Faraday rotation (5 pts)

*Based on material from Jackson.*

Many transparent dielectrics can be made optically active by applying a magnetic field along the axis of light propogation. Consider a vacuum populated by a low-density electron gas and subject to a static magnetic field B_0. For simplicity assume the electrons have no thermal motion. Circularly polarized light is incident on the electron gas. The E-field of the radiation at a fixed point in space is given by

where +/- refers to CW and CCW circular polarization. In steady state, the position of an electron in the gas is described by the vector **r** = β**E**, where β is a constant.

Show that

where ω_c is the cyclotron frequency and +/- refers to CW and CCW circularly polarized light. i.e. The gas has different polarizability (and therefore different ε) depending on the chirality of the circularly polarized light.