Problem 1

(a) An approximate solution of the many-electron Hamiltonian for an atom can be found by neglecting the electron-electron interactions. The resulting Hamiltonian is called the reduced Hamiltonian. Show that a product of atomic orbitals solves the corresponding reduced Hamiltonian.

(b) Because electrons are fermions, there is an additional condition on a many-electron wave function besides being a solution of the Schrödinger equation. Explain this extra condition. How do we ensure that the wavefunction satisfies this condition?

(c) The molecular orbitals of all homonuclear diatomic molecules (H₂, N₂, O₂, etc.) all have a similar structure. Explain what is the same and what is different between the molecular orbitals of the homonuclear diatomic molecules.

(d) How could the H-N-H bond angle be calculated for NH₂?

Problem 2
A two-dimensional crystal has primitive lattice vectors $\vec{a}_1=a\,\hat{x}$ and $\vec{a}_2=2a\,\hat{y}$.

(a) Draw the Wigner-Seitz cell and the first Brillouin zone of this crystal.

(b) Sketch the electron dispersion relation using the empty lattice approximation. There is one valence electron per primitive unit cell, draw the position of the chemical potential in the dispersion relation.

(c) Explain how the electron density of states could be calculated from the dispersion relation you have drawn. Sketch the density of states.

(d) How could the electronic contribution to the specific heat be calculated from the density of states? Sketch the temperature dependence of the specific heat.

Problem 3
A simple approximation for the electron density of an atom is the atomic number times a delta function $Z\delta (\vec{r})$. The atomic number $Z$ is the number of electrons that an atom has. In this approximation, the electron density of a crystal is,

$n(\vec{r}) = \sum \limits_{i,l,m,n}Z_i\delta(\vec{r}_i+l\vec{a}_1+m\vec{a}_2+n\vec{a}_3)$,

where $i$ sums over the atoms in the unit cell and the translation vector $\vec{T}_{lmn}=l\vec{a}_1+m\vec{a}_2+n\vec{a}_3$ repeats the unit cell everywhere in the crystal.

(a) Write down the general expression for a 3-D periodic function in terms of a Fourier series.

(b) What are the atomic form factors for Na ($Z_{Na} = 11$) and Cl ($Z_{Cl} = 17$) in this approximation?

(c) NaCl has an fcc lattice. The reciprocal lattice of fcc is bcc. What are the structure factors $G_{000}$ and $G_{100}$?

Problem 4

(a) The electrical conductivity of a material increases dramatically when light shines on it. What kind of material is this? Does the thermal conductivity also increase dramatically when light shines on it?

(b) How could you experimentally determine the entropy of a bar of iron?

(c) The thermal expansion coefficient is the derivative of the strain with respect to the temperature. How could the thermal expansion coefficient of some material be calculated from the microscopic quantum states of the electrons and phonons? What rank tensor is the thermal expansion coefficient?