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% header
\begin{center}
 {\LARGE {\bf Materials Physics I}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 WS 2007/08\\
 \vspace{8pt}
 {\Large\bf Tutorial 1}
\end{center}

\section{Free electron in a box}
Our understanding of condensed matter is based on the idea of heavy, positively charged ions and light, negatively charged valence electrons.
In order to describe such a system of interacting particles you have to solve the full Hamiltonian
\begin{eqnarray}
 H &=& H_{ion} + H_{el} + H_{ion-el} \nonumber \\
   &=& H_{ion,kin} + H_{ion-ion} + H_{el,kin} + H_{el-el} + H_{ion-el} \nonumber
\end{eqnarray}
which accounts for the ionic and electronic subsystem as well as the coupling between these two.
Obviously, without any approximation this problem is not solvable.

{\bf Born-Oppenheimer} or {\bf adiabatic approximation:}
Lighter valence electrons move much faster than the nuclei and thus follow the ionic motion adiabatically.
For the electrons the nuclei appear fixed in position.
On the other way round the electrons appear blurred to the nuclei adding an extra term to an effective potential.
This approach is basically switching off the interaction of electrons and lattice vibrations.

{\bf Independent electron approximation:}
Having separated the ionic and electronic degrees of freedom the Hamiltonian still involves all electronic coordinates which results in a many-particle wave function as a solution of the Schr"odinger equation depending on the positions of all electrons.
By completely neglecting the electron-electron interaction the idealized Hamiltonian can be written as a sum over single particle Hamiltonians
\[
 H = \sum_i - \frac{\hbar^2}{2m} \nabla_i^2 + v_{ext}({\bf r}_i)
\]
where $v_{ext}$ is the combination of ion-electron and a constant ion-ion interaction.

Using these approximations it is sufficient to consider a single electron located in an effective time-independent potential constituted by the static ions and all other electrons.
Since most materials condense into almost perfect periodic arrays the periodicity should also hold for the potential style.

Within this tutorial even the periodic potential is simplified.
Consider a single particle (mass $m$) enclosed in a box (side length $L=\mathcal{V}^{1/3}$) where the potential is zero inside the box and infinite at the surface.

\begin{enumerate}
 \item Write down the Schr"odinger equation and boundary conditions
       for the particle enclosed in the box.
 \item Find a solution of the Schr"odinger equation.
       Write down the wave function and energy eigenvalues. 
       {\bf Hint:} Apply separation of variables:
       $\Psi({\bf r})=F_x(x)F_y(y)F_z(z)$.
 \item Write down the wave function of the ground state and calculate the
       zero-point energy (energy of the ground state).
 \item What are the values allowed for $k_x$, $k_y$ and $k_z$
       in reciprocal space?
       Sketch a cross section perpendicular to the $k_x$ and $k_y$ axis
       showing some values allowed for $k_x$ and $k_y$.
\end{enumerate}

\section{Reciprocal lattice}

Show that the volume $V_{rec}$ of the unit cell of the reciprocal lattice is given by $V_{rec} = (2\pi)^3/V_{real}$, where $V_{real}$ is the volume of the unit cell in real space.
{\bf Hint:} Use the relation $a \times (b \times c) = b(ac)-c(ab)$ and $a(b \times c)=b(c \times a)$.

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