\pdfoutput=0
\documentclass[a4paper,11pt]{article}
\usepackage[activate]{pdfcprot}
\usepackage{verbatim}
\usepackage{a4}
\usepackage{a4wide}
\usepackage[german]{babel}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{ae}
\usepackage{aecompl}
\usepackage[dvips]{graphicx}
\graphicspath{{./img/}}
\usepackage{color}
\usepackage{pstricks}
\usepackage{pst-node}
\usepackage{rotating}

\setlength{\headheight}{0mm} \setlength{\headsep}{0mm}
\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm}
\setlength{\oddsidemargin}{-10mm}
\setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm}
\setlength{\textheight}{26cm} \setlength{\headsep}{0cm}

\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{\arabic{enumii})}
\renewcommand{\labelenumiii}{\roman{enumiii})}

\begin{document}

% header
\begin{center}
 {\LARGE {\bf Materials Physics II}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 SS 2008\\
 \vspace{8pt}
 {\Large\bf Tutorial 1 - proposed solutions}
\end{center}

\section{Diamagnetism}
\begin{itemize}
 \item Magnetic field ${\bf B}$
 \item Magnetization ${\bf M}$
 \item Suscebtibility $\chi=\frac{\mu_0 {\bf M}}{{\bf B}}$
\end{itemize}

\begin{enumerate}
 \item {\bf Classical approach:}
       \begin{enumerate}
        \item Maxwell: $\oint_{\partial A} E \, ds
	                = -\frac{d}{dt}(\int_A B \, dA)
			\stackrel{B(r)=B}{=}-\frac{d}{dt}(BA)$\\
	      $-\frac{d(BA)}{dt}=-\pi r^2 \dot{B}=U_{ind}$\\
	      $U_{ind}=\oint_{\partial A} E \, ds 
	       \stackrel{E(s)=E}{\Rightarrow}
	       E=-\frac{\pi r^2}{2\pi r}\dot{B}$\\
              $\dot{v}=a=\frac{e}{m}E=-\frac{e}{2m}r\dot{B}
	       \Rightarrow v=-\frac{e}{2m}rB$\\
	      $\omega_L=\frac{v}{r}=-\frac{e}{2m}B$
        \item $I = (\textrm{charge}) \cdot (\textrm{loops per time})
	       \stackrel{1/T=\omega_L/2\pi}{=}
	       (Ze)(\frac{1}{2\pi}\frac{-e}{2m}B)$\\
	      $\mu=IA=I2\pi<\rho^2>=-\frac{Ze^2B}{4m}<\rho^2>$\\
	      $<x^2>=<y^2>=<z^2> \Rightarrow <r^2>=3<x^2>=3<y^2>$\\
	      $<\rho^2>=<x^2>+<y^2>=\frac{2}{3}<r^2>$\\
	      $\mu=-\frac{Ze^2B}{6m}$
        \item $\chi=\frac{\mu_0N\mu}{B}=-\frac{\mu_0NZe^2}{6m}<r^2>$
       \end{enumerate}
 \item {\bf Quantum mechanical theory:}
       \begin{itemize}
        \item vector potential ${\bf A}$
        \item ${\bf B}=\nabla\times{\bf A}$
        \item $
              H_{kin}=\frac{1}{2m}(-i\hbar\nabla_{r}-e{\bf A})^2
              =H_{kin}^0 + H_{kin}'
	      $
       \end{itemize}
       \begin{enumerate}
        \item \begin{eqnarray}
	      H_{kin}&=&\frac{1}{2m}(-\hbar^2\nabla_{r}^2+e^2{\bf A}^2
	                             +i\hbar \nabla_{r}e{\bf A}
			             +e{\bf A}i\hbar \nabla_{r})\nonumber\\
	      H_{kin}^0&=&\frac{-\hbar^2}{2m}\nabla_r^2\nonumber\\
	      H_{kin}'&=&\frac{i\hbar e}{2m}(\nabla_r{\bf A}+{\bf A}\nabla_r)+
	                 \frac{e^2{\bf A}^2}{2m}\nonumber
	      \end{eqnarray}
	      Terms in $H_{kin}'$ can be treated as small perturbation.
        \item ${\bf A}=\left(-\frac{1}{2}yB,\frac{1}{2}xB,0\right)$, since:
	      $\nabla_r\times{\bf A}=\left(0,0,\frac{1}{2}B+\frac{1}{2}B\right)=
	       \left(0,0,B\right)$\\
	      Note: $\nabla_r{\bf A}=0$
        \item $
	      H_{kin}'=\frac{i\hbar e}{2m}\left(
	       x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}
	                                  \right)+\frac{e^2B^2}{8m}(x^2+y^2)
	      $\\
	      $
	       L_z=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}
	       \Rightarrow  H_{kin}'=\frac{i\hbar e}{2m}L_z+
	       \frac{e^2B^2}{8m}(x^2+y^2)
	      $
        \item $\chi=-\frac{1}{V}\mu_0\frac{\partial^2 E}{\partial B^2}
	       \Rightarrow$
	      only second term contributes to $\chi$!\\
	      $\chi=-\frac{1}{V}\mu_0\frac{e^2}{4m}<\phi|(x^2+y^2)|\phi>$
	\item $<\phi|x^2|\phi>=<\phi|y^2|\phi>=\frac{1}{3}<\phi|r^2|\phi>$\\
	      $\Rightarrow \chi=-\frac{1}{V}\mu_0\frac{e^2}{6m}
	       <\phi|r^2|\phi>$\\
	      Consider all $Z$ electrons and all atoms per volume:\\
              $\chi=-\frac{\mu_0NZe^2}{6m}<\phi|r^2|\phi>$
       \end{enumerate}
\end{enumerate}

\end{document}
