\section*{Problems}

\begin{enumerate}
\renewcommand{\labelenumi}{ [\arabic{chapter}-\arabic{enumi}] }

\item Using Eq.\,(\ref{eq-Henergy}), estimate the 1s energy of B, C and
N atoms.  In order to observe the core state energy, we usually use
XPS (X ray photo-electron spectroscopy) measurements. Explain how to
measure 1s states for these atomic species by XPS with some
illustrative figures.

\item An electronic $p$ orbital has angular momentum $\ell =
1$. Obtain the spherical harmonic $Y_{\ell m}(\theta,\phi)$ for $\ell
= 1$ and $m = -1, 0, 1$. By combining the three functions, construct
the $p_x$, $p_y$ and $p_z$ functions. Plot the shape of the $p_x$,
$p_y$ and $p_z$ functions in three dimensions.

\item Plot the rough shape of $R_{n\ell}(r)$ for 1s, 2s and 2p states as
a function of $r$.  Explain how these functions are orthogonal to each
other.

\item When we change $Z$ to $Z-2$ in Eq.\,(\ref{eq-Henergy}) for
expressing the screening between two 1s electrons, estimate the 2s
energy of B, C and N atoms.

\item In order to solve Eq.\,(2.1), express the
wavefunction as $\Psi({\bf r})=R(r)\Theta(\theta)\Phi(\phi)$ and
obtain the equations for the variables of $r$, $\theta$, and
$\phi$. Explain qualitatively why $2p$ orbitals have a higher energy
than $2s$ orbitals. It is only for the case of the hydrogen atom that
the $2s$ and $2p$ levels have an identical energy.


\item Using the unitary matrix $U$ in Eq.\,(2.16), diagonalize
the Hamiltonian in Eq.\,(2.14).

\item Solve the Schr\"odinger equation for the H molecule for the case that
$s=\langle\Psi_1|\Psi_2\rangle$ is not zero.
Obtain both eigenvalues and wavefunctions.

\item Use the tight binding model to obtain the electronic band
structure for $\pi$ electrons in polyacetylene. In the case of
polyacetylene, there are two kinds of structures, the cis and trans
structures (see Fig.\,(\ref{f-polyactrans}). In the case of
trans-polyacetylene, the transfer integral $t$ for $\pi$ electrons
should have two values $t_1$ and $t_2$ depending on the double and
single bond nature. Show that an energy gap which is proportional to
$|t_1-t_2|$ appears in the case of trans-polyacetylene.
\begin{figure}
\includegraphics[width=8cm]{./eps/polyacetinele-trans.eps}
\caption[]{The structure of trans-polyacetylene, with alternative C-C bonds
in the {\it trans} configuration. In cis-polyactylene (not shown), an
 armchair edge shaped C-C chain appears. }
 \label{f-polyactrans}
\end{figure}


\item Obtain the coordinates in $k$ space for the K, M, and $\Gamma$
points and obtain the energy values for the simplest tight binding energy
for graphene
at these high symmetry points in the Brillouin zone (Eq.\,(\ref{eq:gre})).

\item Obtain Eqs.\,(\ref{eq:gre-wall}) and (\ref{eq:gre-wall-v}). Evaluate the value of the Fermi velocity. 

\item Show that the density of states is proportional to $E$ when the
energy dispersion is $E = a \sqrt{k_x^2 + k_y^2}$
in two dimensional materials. How about for the
linear dispersion in one-dimensional materials?

\item Plot the density of states for the parabolic energy band, $E = a
(k_x^2 + k_y^2)$.

\item By expanding $f(k)$ of Eq.\,(\ref{eq:fk}) near the K point, 
show that the Hamiltonian matrix is written by Eq.\,(\ref{eq-diraclike}). 


\item Explain how to obtain the density of states for the cosine
energy band, $E = a ( \cos(k_x a) + \cos(k_y a) )$ numerically.

\item Calculate the diameter for an $(n,m)$ SWNT. What are the values
of diameters for (5,5), (9,0), (10,5) SWNTs. What are possible $(n,m)$
values for SWNTs having a  1.5nm diameter?

\item For a given $(n,m)$ SWNTs, show the expression of $\vec{T} =
(t_1, t_2)$ as a function of $n, m$. What is the length of $\vec{T}$?

\item Show that $\left|\vec a_1\times\vec a_2\right|=\sqrt{3}a^2/2$
and that the number of hexagons in the nanotube unit cell is
$N=2\left(n^2+nm+m^2\right)/d_R$.

\item Show the relation for an $(n,m)$ SWNT $\left|\vec
C_h\times\vec T\right|=\sqrt{3}a^2\left(n^2+nm+m^2\right)/d_R$, where
$d_R={\rm gcm}(2n+m, 2m+n)$ and gcm is an integer function of the greatest
common multiplier.

\item Show the reciprocal lattice unit vectors $\left(\vec K_1,\vec
K_2\right)$ as a function of $n, m$ in 2D reciprocal space.

\item Show that $\left|\vec K_1\right|=2/d_t$ and $\left|\vec
K_2\right|=2\pi/\left|\vec T\right|$.

\item Give the 1D electronic structure for the $(10,10)$
and $(18,0)$ SWNTs. Plot the cutting lines (1D Brillouin zone of SWNTs)
in the 2D reciprocal space of the 2D Brillouin zone of graphene for these
two SWNTs.

\item Show that in Fig.\,(\ref{f21-7},
$\stackrel{\longrightarrow}{YK} = [(2n+m)/{3}]{\bf
K}_1$ in Eq.\,(\ref{eq:YK}).

\item Plot the cutting lines for type-I, type-II semiconducting and
metallic SWNTs and show which cutting lines correspond to $E_{11}^S$,
$E_{22}^S$, $E_{33}^S$, and $E_{11}^M$    transitions for each case.

\end{enumerate}
