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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 - proposed solutions}
\end{center}

\section{Free electron in a box}
\begin{enumerate}
 \item

 \begin{itemize}
  \item Schr"odinger equation:\\
        \[
	- \frac{\hbar^2}{2m} \nabla^2 \Psi({\bf r}) + V({\bf r}) \Psi({\bf r})
	= E \Psi({\bf r}) \textrm{ , }
	V({\bf r}) = 0 \textrm{ for } {\bf r} \in [0,L]
	\]
  \item Boundary conditions: $\Psi({\bf r}) = \Psi(x,y,z)$\\
        \[
        \Psi(0,y,z) = \Psi(L,y,z) = 0 \qquad
        \Psi(x,0,z) = \Psi(x,L,z) = 0 \qquad
        \Psi(x,y,0) = \Psi(x,y,L) = 0
	\]
 \end{itemize}


 \item
  \begin{itemize}
   \item Product ansatz: $\Psi({\bf r})=F_x(x)F_y(y)F_z(z)$, with\\
         $F_x(x)=0$ for $x=0,L$\\
         $F_y(y)=0$ for $y=0,L$\\
         $F_z(z)=0$ for $z=0,L$.
   \item Schr"odinger equation:\\
         Use: $\nabla^2=\frac{\partial^2}{\partial x^2} +
                        \frac{\partial^2}{\partial y^2} +
                        \frac{\partial^2}{\partial z^2} \Rightarrow$\\
	 \[
	 - \frac{\hbar^2}{2m} \Big[
	 F_y(y) F_z(z) \frac{d^2}{dx^2} F_x(x) +
	 F_x(x) F_z(z) \frac{d^2}{dy^2} F_y(y) +
	 F_x(x) F_y(y) \frac{d^2}{dz^2} F_z(z)
	 \Big] =
	 E F_x(x) F_y(y) F_z(z)
	 \]
   \item Schr"odinger equation fullfilled if:
         \[
         - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} F_x(x) = E_x F_x(x), \quad
         - \frac{\hbar^2}{2m} \frac{d^2}{dy^2} F_y(y) = E_y F_y(y),\quad
         - \frac{\hbar^2}{2m} \frac{d^2}{dz^2} F_z(z) = E_x F_z(z).
	 \]
	 \[
         \Rightarrow \Big[E_x + E_y + E_z\Big] F_x(x) F_y(y) F_z(z) =
	             E F_x(x)F_y(y)F_z(z) \textrm{, } \quad E = E_x + E_y + E_z
	 \]
         Three eigenvalue problems of the same character.
	 Sufficient to examine only one!
   \item Solution of the Schr"odinger equation:\\
         \begin{itemize}
	  \item Ansatz: $F_x = A_x \exp(ik_xx) + B_x \exp(-ik_xx)$
	  \item Boundary conditions:\\
	        $F_x(0)=0 \Rightarrow B_x=-A_x$, \quad
		let ${A}_x = \frac{1}{2i}\tilde{A}_x$
		$\Rightarrow$ $F_x(x) = \tilde{A}_x \sin(k_xx)$\\
		$F_x(L)=0 \Rightarrow k_x L = n_x \pi$ or rather
		$k_x=n_x \frac{\pi}{L}$, \quad $n_x=0,\pm1,\pm2,\ldots$
          \item Forbidden values for $n_x$:\\
	        $n_x \ne 0$: otherwise wave function zero for all $x$\\
	        $n_x > 0$: wave functions for $+n_x$ and $-n_x$
		not linearly independent (same quantum sate)
         \end{itemize}
	 \[
	 \Rightarrow
	 \Psi_{n_x n_y n_z} = A \sin(\frac{n_x \pi}{L}x)
	                        \sin(\frac{n_y \pi}{L}y)
				\sin(\frac{n_z \pi}{L}z),
	 \quad A=\tilde{A}_x\tilde{A}_y\tilde{A}_z
	 \]
   \item Energy\\
         \[
	 E_{n_x n_y n_z} = \frac{\hbar^2 \pi^2}{2m L^2}(n_x^2+n_y^2+n_z^2)
	 \]
  \end{itemize}

 \item Ground-state:
       \[
       \Psi_{111} = A \sin(\frac{\pi}{L}x) \sin(\frac{\pi}{L}x)
                      \sin(\frac{\pi}{L}x) \qquad
       E_{111} = \frac{\hbar^2 \pi^2}{2m L^2} (1+1+1)
                = \frac{3 \hbar^2 \pi^2}{2m L^2}
       \]
 \item $n_x,n_y,n_z=1,2,3\ldots$\\
       Allowed $k_{x,y,z}$ values located in positive octant only.
       \begin{center}
       \includegraphics[width=10cm]{feg_kvals.eps}
       \end{center}

\end{enumerate}

\section{Reciprocal lattice}

Convention:
\begin{itemize}
 \item basis of unit cell in real space: $a_1,a_2,a_3$
 \item basis of unit cell in reciprocal space: $b_1,b_2,b_3$
\end{itemize}
Prove:
\[
V_{real}=a_1(a_2 \times a_3)
\]
\[
V_{rec}=b_1 ( b_2 \times b_3)
       =\frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3} (a_2 \times a_3) [
       (a_3 \times a_1) \times (a_1 \times a_2) ]
\]
\[
\textrm{hint 1: }
(a_3 \times a_1) \times (a_1 \times a_2) =
a_1((a_3 \times a_1)a_2) - a_2((a_3 \times a_1)a_1) =
a_1((a_3 \times a_1)a_2)
\]
\[
\Rightarrow V_{rec}= \frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3} 
(a_2 \times a_3) (a_1(a_3 \times a_1) a_2)
\]

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