\section*{Problems}

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

\item When we consider the family of $(n,m)$ nanotubes, there are two
definitions. One is for the mod$(2n+m,3)$ family, where
mod($x,y$) is an integer function of the modulus
(i.e. mod(20,7)=6) and another definition is for the mod$(n-m,3)$
family.  Obtain the relationship between the
mod$(2n+m,3)$ family and the mod$(n-m,3)$ family.

\item When the planar sp$^2$ bond is bend in the circumferential direction with an angle $\theta$ which is on the order of $\theta=0.1$rad,
we expect a large curvature effect. What is
corresponding tube diameter?

\item The curvature effect can be understood by the Slater--Koster method
 in which the transfer matrix element for
 $\pi$-$\pi$ orbitals is mixed with $\sigma$-$\sigma$ orbitals.
 Show how the matrix element is modified as a function of $\theta$
in Problem [10-2].

\item Applying the wavefunction of Eq.\,(\ref{eq-Hexcitonwave}) to
Eq.\,(\ref{eq-Hexciton}), obtain the differential equations for $g$ and $f$.

\item Solve the one-dimensional hydrogen Schr\"odinger equation.  In
 particular show that the lowest energy is minus infinity. Obtain the
 corresponding wavefunction.

\item Solve the two dimensional hydrogen Schr\"odinger equation.
In this case, we have an angular momentum within a plane. Obtain the energies.

\item When the potential is spherical in three dimensions,
the potential is given as a function of $r$. In this case, show that the
angular part of the wavefunction for the Hamiltonian with a  spherical
potential is given by the spherical harmonics, $Y_{lm}(\theta, \varphi)$.

\item Obtain the differential equation on $r$ for a hydrogen atom with angular
momentum $\ell$, and solve this equation for the bound states of the
hydrogen atom..

\item Evaluate the 1s exciton binding energy
of Si by investigating the dielectric constant and
effective masses for an electron and a hole.
What is the corresponding temperature?

\item Explain what are the direct and indirect energy gaps by showing
some example in the case of semiconductors. In the case of an indirect
energy gap, explain that the exciton can not emit a photon.

\item When electron-electron interaction $U$ exists between two electrons,
explain that the interaction modifies electronic
states mainly near the Fermi energy. In
particular, why are the electrons at the bottom of
the energy band not affected much by the Coulomb interaction.
Consider the two cases that (1) $U >> W$ ($W$: energy band width),
and (2) $W >> U$.

\item Explain qualitatively what you expect for the energy dispersion
$E(k)$, if electrons have a finite life time in the presence of some
interaction.

\item When the wavefunction is spatially localized, the wavefunction
is expressed by a linear combination of Bloch
wavefunctions with many $k$ values. In order to understand this, calculate
the Fourier transform of a Gaussian function $f(x) = \exp(-x^2/a^2)$.

\item The spin functions of an electron with $s_z= 1/2$ and $-1/2$
are denoted by $\alpha$ and $\beta$,
respectively. When we define the spin of an hole by
$s_z= -1/2$ and $1/2$, respectively, for the
states which are originally occupied by an
electron with $s_z= 1/2$ and $-1/2$ and denoted by
$\beta_h$ and $\alpha_h$, obtain the total spin function of a singlet
($S=0$) and a triplet ($S=1$) exciton.

\item When we consider the electric-dipole transition which does not
depend on spin, explain that the $S=1$, $S_z = -1, 0, 1$ exciton
states are all dark exciton states.

\item In the presence of a Coulomb interaction, an energy difference
between the $S=1$ and $S=0$ states of two
electrons appears, which is known as the exchange
energy.  Please explain the physical meaning of
the exchange energy in terms of Pauli exclusion
principles, Hartree-Fock approximation, and spin
of two electrons. Why are the $S=1$ states lower
in energy than the $S=0$ states?

\item Explain the reason why the $S=0$ exciton does not relax quickly
to the lower energy $S=1$ states.

\item Derive Eq.\,(\ref{eq:total}) for achiral SWNTs and draw
schematics similar to those of Fig.\,\ref{eband} for both zigzag and
armchair tubes.

\item Illustrate figures similar to Fig.\,\ref{bz} for the $E_{11}$ exciton
levels for $A_1$, $A_2$ and $E$ symmetries. What happens for
 $E_{22}$ or $E_{12}$ excitons? Note that two
 configurations are coupled when explaining $A_1$ and
 $A_2$ excitons (see Eq.\,(\ref{A21})).

\item Why does the plus sign in Eq.\,(\ref{A21}) correspond to the
 anti-symmetric $A_2$ exciton? Discuss how the wavefunctions of the
 valence and conduction bands change by a $C_2$ rotation.

\item Compare Eq.\,(\ref{log}) with the logarithmic part of Eq.\,(\ref{eq1-f})
and discuss the results.

\item Show that the effective mass for the $E_{33}^S$ or $E_{44}^S$
exciton is larger than that for $E_{11}^S$ or $E_{22}^S$.

\item By considering the 1s energy of a hydrogen atom, show that
the exciton binding energy becomes large when the effective mass
of the electron and hole is large. From this, show that the exciton binding
energy of the $E_{33}^S$ or $E_{44}^S$ exciton is larger that for
the $E_{11}^S$ or $E_{22}^S$ exciton.

\item When the dielectric constant of the surrounding material
$\kappa_{\rm env}$ is large, which exciton feels $\kappa_{\rm
env}$ more, $E_{11}^S$ and $E_{33}^S$? Explain by illustrating the
electric field for these excitons.

\item In a $2n+m=$const family, which SWNTs have a larger
effective mass, near armchair SWNTs or near zigzag SWNTs?

\item Using the fact that the distance of the K point from the $\Gamma$
point in the direction perpendicular to the cutting line is,
$K\Gamma=(2n+m)/3$, illustrate the cutting lines for $E_{11}^S$ to
$E_{44}^S$ for S1 and S2 SWNTs in the two-dimensional Brillouin
zone.

\item For $E_{11}^S$, which semiconducting type does has a larger effective
mass, S1 or S2? How about for $E_{22}^S$?


\end{enumerate}
