\pdfoutput=0
-\documentclass[twoside,a4paper,11pt]{book}
+\documentclass[twoside,a4paper,11pt,openany]{book}
%\documentclass[twoside,a4paper,11pt,draft]{book}
\usepackage[activate]{pdfcprot}
\usepackage{verbatim}
\include{qm}
\include{stat}
\include{solid}
+\include{sim}
\appendix{}
-\include{app}
+\part{Appendices}
+\include{math}
\backmatter{}
\include{literature}
\section{Variational method}
\label{sec:var_meth}
+The variational method constitutes a promising approach to estimate the ground-state energy $E_0$ of a system for which exact solutions are unknown.
+Considering a {\em trial ket} $|\tilde 0\rangle$, which tries to imitate the true ground-state ket $|0\rangle$, it can be shown that
+\begin{equation}
+\tilde E\equiv\frac{\langle \tilde 0|H|\tilde 0\rangle}{\langle \tilde 0|\tilde 0\rangle}
+\ge E_0 \textrm{ ,}
+\end{equation}
+i.e.\ an upper bound to the ground-state energy can be obtained by considering various kinds of $|\tilde 0\rangle$.
+To proof this, $|\tilde 0\rangle$ is expanded by the exact energy eigenkets $|k\rangle$ with
+\begin{equation}
+H|k\rangle = E_k|k\rangle\text{ ,}
+\qquad E_0\leq E_1\leq\ldots\leq E_k\ldots \text{ ,}
+\qquad \langle k|k'\rangle=\delta_{k k'} \text{ ,}
+\label{sec:vm_d}
+\end{equation}
+which are unknown but, still, form a complete and orthonormal basis set, to read
+\begin{equation}
+|\tilde 0\rangle = \vec{1} |\tilde 0\rangle
+ = \sum_{k=0}^{\infty} |k\rangle\langle k|\tilde 0\rangle
+\text{ .}
+\end{equation}
+Since $\langle k|k'\rangle=\delta_{k k'}$, $H|k\rangle = E_k|k\rangle$ and $E_k\geq E_0$ (see \eqref{sec:vm_d})
+\begin{equation}
+\tilde E=
+\frac{\sum_{k,k'}\langle \tilde 0|k\rangle\langle k|H|k'\rangle\langle k'|\tilde 0\rangle}
+ {\sum_{k,k'}\langle \tilde 0|k\rangle\langle k|k'\rangle\langle k'|\tilde 0\rangle}=
+\frac{\sum_k \left| \langle k|\tilde 0\rangle \right|^2 E_k}
+ {\sum_k \left| \langle k|\tilde 0\rangle \right|^2} \geq
+\frac{\sum_k \left| \langle k|\tilde 0\rangle \right|^2 E_0}
+ {\sum_k \left| \langle k|\tilde 0\rangle \right|^2}=E_0
+\text{ ,}
+\label{sec:vm_f}
+\end{equation}
+which proofs the variational theorem.
+Moreover, equality in \eqref{sec:vm_f} is only achieved if $|\tilde 0\rangle$ coincides exactly with $|0\rangle$, i.e.\ if the coefficients $\langle k|\tilde 0\rangle$ all vanish for $k\neq 0$.
+
\chapter{Quantum dynamics}
\subsubsection{Hohenberg-Kohn theorem}
-Considering a system with a nondegenerate ground state, there is obviously only one ground-state charge density $n_0(\vec{r})$ that correpsonds to a given potential $V(\vec{r})$.
+Considering a system with a nondegenerate ground state, there is obviously only one ground-state charge density $n_0(\vec{r})$ that corresponds to a given potential $V(\vec{r})$.
In 1964, Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}.
For a nondegenerate ground state, the ground-state charge density uniquely determines the external potential in which the electrons reside.
The proof presented by Hohenberg and Kohn proceeds by {\em reductio ad absurdum}.
\int n(\vec{r}) \left( V_2(\vec{r})-V_1(\vec{r}) \right) d\vec{r}
}_{=0}
\end{equation}
-is revealed.
+is revealed, which proofs the Hohenberg Kohn theorem.
projects / lectures / latex.git / commitdiff