\section{Problems}

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

\item Using Maxwell's equations, explain why we need a gauge field
for the vector potential {\bf A} and a static potential
$\phi$. Consider some gauges explicitly and explain under what kind of
situation such gauges are useful by giving some explanation
of the physical phenomena.

\item In vacuum, show that the electric field is expressed by the
vector potential {\bf A}.


\item When we consider the Hamiltonian in the presence of a vector
potential, expand the Hamiltonian and retain the linear term in {\bf
A}.  This corresponds to a perturbation Hamiltonian of the
electron-photon coupling constant. Use the Coulomb gauge div{\bf A}=0
when you obtain this result.

\item In the previous problem, we also have a term which is
proportional to ${\bf A}^2$. In order to neglect this term, this term
should be at least 1/10 smaller than the linear {\bf A} term. What is
the corresponding value of the electric field? If {\bf A} gives an
electric field above this value, we should then consider the
non-linear ${\bf A}^2$ effect of light.\footnote{The general
non-linear optical effect in materials occurs at a much lower power
level for light since the physical properties are saturated as a
function of $E$. For example, the polarization vector ${\bf P}$ can be
expanded as $\alpha E + \alpha(2) E^2 + \ldots$ etc. }

\item The Poynting vector (note that it is not the Pointing vector), ${\bf
S} = {\bf E} \times {\bf H}$ is the power density per unit area of
the electromagnetic field. In a typical micro-Raman measurement system, the
diameter of the light beam is about 1$\mu$m and the laser power is
1\,mW. Estimate the power density of this micro-Raman setup and
calculate ${\bf E}$. Show that the electric field thus obtained is not
strong enough to be in the non-linear regime.

\item Show that there is $C_2$ rotational symmetry in graphene. $C_2$
rotational symmetry means that the lattice structure does not change
under a 180$^{\circ}$ rotation about a point.  Specify the symmetry axis of
this $C_2$ rotation.

\item By the $C_2$ rotation, the  A and B carbon atoms in the unit cell are
exchanged with each other. Show that all A and B carbon atoms in the lattice are exchanged for any $C_2$ rotation.

\item When we put the origin of the coordinate system at the axis
point of the $C_2$ rotation of graphene, show that the perturbation
Hamiltonian for the interaction of graphene with an incident light
beam is an odd function of the coordinates.

\item When we solve a simple $2\times 2$ tight binding Hamiltonian for the
$\pi$ band of graphene (Eq.\,(\ref{eq:SH-g})), we get the wavefunction
$\Psi({\bf k})$ as a linear combination of the Bloch functions
$\Phi_A({\bf k})$ and $\Phi_B({\bf k})$ consisting of A and B terms:
$$
\Psi({\bf k}) = C_A({\bf k}) \Phi_A({\bf k}) + C_B({\bf k}) \Phi_B({\bf k}) .
$$
Obtain the explicit form of $C_A({\bf k})$ and $C_B({\bf k})$ for
$s=0$ in Eq.\,(\ref{eq:SH-g}) with the use of $f(k)$ and $w(k)$ as defined in
Eqs.\,(\ref{eq:fk}) and (\ref{eq:wk}), respectively.

\item In the calculation for problem [5-9], show $C_A = C_B$ and $C_A = -C_B$
for the valence and conduction band, respectively, for any ${\bf k}$
point. Combining this result
with the fact that the perturbation Hamiltonian
is an odd function of the coordinates
for exchanging A and B, obtain the dipole transition matrix element
which is proportional $\langle \pi^* | \nabla | \pi \rangle$ as a
function of ${\bf k}$.

\item Using time dependent perturbation theory, obtain the result for
$|a_{m}^{(1)}(t)|^2$ in Eq.\,(\ref{e4.26y}).

\item Plot $|a_{m}^{(1)}(t)|^2$ in Eq.\,(\ref{e4.26y}) for the case of
$\omega_{m\ell} = \omega/2$, $2\omega/3$, $3\omega/4$, and $\omega$.
Show that the peak height of the oscillation is increasing as
$\omega_{m\ell}$ increases. This means that we
can select only the closest $m$ states for the calculation of $|a^{(1)}(t)|^2$
as a first approximation.

\item A delta function $\delta(x)$ has two significant properties;
(1) The values of $\delta(x)=\infty$ for $x=0$, but =0 for $x\ne 0$.
(2) When we integrate $\delta(x)$ over any region which includes $x=0$,
 the integrated value is 1. Using these definitions,
obtain the following formula;
$$
\lim_{t\to\infty} \frac{\sin^2(\alpha t)}{\pi\alpha^2 t} = \delta(\alpha) ,
$$
and then using this formula, obtain the Fermi Golden Rule directly.

\item The uncertainty relation of $\Delta E \Delta t \sim \hbar$
(Eq.\,(\ref{eqEt})) is important in spectroscopy since most optical
processes have a finite life time. If an excited state has a life time
$\Delta t$, the energy of the excited states has an energy uncertainty value
$\Delta E$. In this case, time-dependent theory in which
we consider a definite energy, should be modified. Explain how the
resonance condition can be relaxed in the case of such a life time.

\item For a photo-excited electron in a carbon nanotube, the electron
can emit a phonon within 1\,ps. Estimate the uncertainty energy value
for this electron. For the photoluminescence process for which a
photon is emitted, the excited electron has a relatively slower life
time, on the order of 1\,ns. Is the accuracy of the spectrometer
sufficient to observe the energy of a photoluminescent phonon?

\item Analyze the first-order processes depicted in the Feynman
diagrams of Fig.\,\ref{ff10.10} within the framework of a resonant process,
which is obtained by having null terms in Eq.\,(\ref{ee10.58}).  By
starting with a given $\hbar \omega_i$ and the material in the ground
state, which processes can be and which cannot be resonant? Consider
$n = n'$ for simplicity, which happens when the phonon does not
break the symmetry of the system.

\item In the previous problem, derive a quantitative analysis to
understand which of the six processes in Eq.\,(\ref{ee10.58}) dominate
the total intensity. Choose values for $E_m-E_i$ and for
$\hbar\omega_q$ and introduce the damping term $i\gamma_r$ with a specific
value for $\gamma_r$.  You will realize that processes which seem
"reasonable" within our own "common sense" are closely related to
whether or not the resonance condition is achieved.

\item When we consider one optical-absorption, and one optical-emission,
and one phonon-emission vertex, we can get 6 possible Feynman diagrams.
How many Feynman diagrams are expected if we change one phonon-emission
vertex into a two phonon-emission vertex?  Illustrate using some
diagrams which process starts from the optical absorption.

\item Show that, for a periodic motion of charged particles
forming a dipole $d$, and $c$ is the velocity of light,
the intensity of radiation with frequency $\omega$ is given by
\begin{equation}
I = \frac{4\omega^2}{3c^3}|d|^2.
\label{eq-dipole}
\end{equation}


\end{enumerate}

