From: hackbard <hackbard@sage.physik.uni-augsburg.de>
Date: Fri, 19 Oct 2007 14:48:36 +0000 (+0200)
Subject: added solid state physics tutorial 1
X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=commitdiff_plain;h=8e0003a7f7b1cb5bb75f5037388847cf4579e711

added solid state physics tutorial 1
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diff --git a/solid_state_physics/tutorial/1_01.tex b/solid_state_physics/tutorial/1_01.tex
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+\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})}
+
+\begin{document}
+
+% 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.
+
+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 is called the Born-Oppenheimer or adiabatic approximation basically switching off electron-phonon interactions.
+
+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 Hamiltonian can be written as a sum of 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.
+This is called the independent electron approximation.
+
+Thus, it is sufficient to consider a single electron located in an effective time-independent potential due to 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 $L=V^{1/3}$) where the potential is constant ($V_0$) 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}
+
+\end{document}