Problem 1
A computer program is used to calculate the molecular orbitals of methane (CH4). The resulting molecular orbitals $\phi_1(\vec{r}),\cdots,\phi_N(\vec{r})$ have the energies $E_1,\cdots,E_N$. Carbon has 6 electrons and hydrogen has one. What is the multi-electron ground state wavefunction and what is multi-electron wavefunction of the first excited state? What is the approximate energy difference between the ground state and the first excited state?
Problem 2
The phonon eigen functions are also eigen functions of the translation operator. What are these eigen functions for a bcc lattice? Give the eigen functions in Cartesian coordinates $(x, y, z)$.
The primitive lattice vectors in real space are:
\[ \begin{equation} \large \vec{a}_1= \frac{a}{2}(\hat{x}+\hat{y}-\hat{z}), \,\, \vec{a}_2= \frac{a}{2}(-\hat{x}+\hat{y}+\hat{z}), \,\, \vec{a}_3= \frac{a}{2}(\hat{x}-\hat{y}+\hat{z}) \end{equation} \]Draw the phonon dispersion curves for a bcc crystal with one atom in the basis along $N$ - $\Gamma$ - $P$ - $H$.
\[ \begin{equation} \large \overline{\Gamma N}= \frac{\sqrt{2}\pi}{a},\hspace{0.3cm}\overline{\Gamma P}= \frac{\sqrt{3}\pi}{a},\hspace{0.3cm}\overline{\Gamma H}= \frac{2\pi}{a} \end{equation} \]Problem 3
In the free electron gas model for metals, explain
(a) how the electron density can be measured experimentally.
(b) how the electrical conductivity is related to the thermal conductivity.
Problem 4
An indirect bandgap semiconductor has a valence band maximum at $\vec{k}=0$ and a conduction band minimum at $\vec{k}_{\text{min}}= \frac{\pi}{3a}\hat{k}_x$ where $a = 0.5$ nm. The effective mass in the conduction band is equal to the free electron mass, $m^*=$ 9.11 × 10-31 kg. What is the energy of an electron in the conduction band with a $k$-vector $\vec{k}_{\text{min}}+ \frac{\pi}{10a}\hat{k}_y$?
$\hbar = 1.05\times 10^{-34}\,\text{J s}$