solid_state_physics/tutorial/1_04s.tex

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  28 \begin{document}
  29
  30 % header
  31 \begin{center}
  32  {\LARGE {\bf Materials Physics I}\\}
  33  \vspace{8pt}
  34  Prof. B. Stritzker\\
  35  WS 2007/08\\
  36  \vspace{8pt}
  37  {\Large\bf Tutorial 4 - proposed solutions}
  38 \end{center}
  39
  40 \section{Hall effect and magnetoresistance}
  41 \begin{enumerate}
  42  \item \begin{itemize}
  43         \item probability: $1-\frac{dt}{\tau}$
  44         \item momentum contribution of non-colliding electrons:
  45               $f(t)dt+O(dt)^2$
  46         \item momentum per electron at time $t+dt$:\\
  47               \[
  48               p(t+dt)=\left(1-\frac{dt}{\tau}\right)
  49                       \left[p(t)+f(t)dt+O(dt)^2\right]
  50                      =p(t)-\frac{dt}{\tau}p(t)+f(t)+O(dt)^2
  51               \]
  52        \end{itemize}
  53  \item \[
  54        p(t+dt)-p(t)=-\frac{dt}{\tau}p(t)+f(t)dt+O(dt)^2
  55        \]
  56        \[
  57        \frac{p(t+dt)-p(t)}{dt}=-\frac{p(t)}{\tau}+f(t)+\frac{O(dt)^2}{dt}
  58        \]
  59        \[
  60        \stackrel{dt\rightarrow 0}{\Rightarrow} \quad
  61        \frac{dp(t)}{dt}=-\frac{p(t)}{\tau}+f(t)
  62        \]
  63  \item
  64 \end{enumerate}
  65
  66 \end{document}