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

(A&M Eq. 27.43)

where the restoring force on the the electron is described by a spring constant K = mω_0^2.

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

(2)

  • 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 = βE, 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.