From: hackbard Date: Wed, 15 Feb 2012 13:30:32 +0000 (+0100) Subject: theorems and hyperrefs X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=4069f849b1973e2eb9020fa2e190f93b56537c60;p=lectures%2Flatex.git theorems and hyperrefs --- diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex index 5f2b0ca..f1f0ae3 100644 --- a/physics_compact/math_app.tex +++ b/physics_compact/math_app.tex @@ -5,7 +5,7 @@ \subsection{Vector space} \label{math_app:vector_space} -\begin{definition} +\begin{definition}[Vector space] A vector space $V$ over a field $(K,+,\cdot)$ is an additive abelian group $(V,+)$ and an additionally defined scalar multiplication of $\vec{v}\in V$ by $\lambda\in K$, which fullfills: \begin{itemize} \item $\forall \vec{v} \, \exists 1$ with: $\vec{v}1=\vec{v}$ @@ -36,7 +36,7 @@ The addition of two vectors is called vector addition. \subsection{Dual space} -\begin{definition} +\begin{definition}[Dual space] The dual space $V^{\dagger}$ of vector space $V$ over field $K$ is defined as the set of all linear maps from the vector space $V$ into its field $K$ \begin{equation} \varphi:V\rightarrow K \text{ .} @@ -55,13 +55,9 @@ The map $V^{\dagger}\times V \rightarrow K: [\varphi,\vec{v}]=\varphi(\vec{v})$ \subsection{Inner and outer product} \label{math_app:product} -\begin{definition} +\begin{definition}[Inner product] The inner product on a vector space $V$ over $K$ is a map -\begin{equation} -(\cdot,\cdot):V\times V \rightarrow K -\text{ ,} -\end{equation} -which satisfies +$(\cdot,\cdot):V\times V \rightarrow K$, which satisfies \begin{itemize} \item $(\vec{u},\vec{v})=(\vec{v},\vec{u})^*$ (conjugate symmetry, symmetric for $K=\mathbb{R}$) @@ -117,7 +113,7 @@ or the conjugate transpose in matrix formalism In doing so, the conjugate transpose is associated with the dual vector. \end{remark} -\begin{definition} +\begin{definition}[Outer product] If $\vec{u}\in U$, $\vec{v}\in V$ are vectors within the respective vector spaces and $\vec{\varphi}\in V^{\dagger}$ is a linear functional of the dual space $V^{\dagger}$ of $V$, the outer product $\vec{u}\otimes\vec{v}$ is defined as the tensor product of $\vec{\varphi}$ and $\vec{u}$, which constitutes a map $A:V\rightarrow U$ by diff --git a/physics_compact/phys_comp.tex b/physics_compact/phys_comp.tex index c82c627..8a9760a 100644 --- a/physics_compact/phys_comp.tex +++ b/physics_compact/phys_comp.tex @@ -46,6 +46,13 @@ % units \usepackage{units} +% theorem environment +\usepackage{framed} +\usepackage[framed,thref,amsmath,hyperref]{ntheorem} + +% hyperlinks +\usepackage[colorlinks=true,linkcolor=black,citecolor=black]{hyperref} + % shortcuts \newcommand{\si}{Si$_{\text{i}}${}} \newcommand{\ci}{C$_{\text{i}}${}} @@ -87,6 +94,11 @@ \newrgbcolor{hb}{0.75 0.77 0.89} \newrgbcolor{lbb}{0.75 0.8 0.88} \newrgbcolor{lachs}{1.0 .93 .81} +% +\newrgbcolor{ctheorem}{0.9 0.9 0.95} +\newrgbcolor{cdefinition}{0.95 0.9 0.9} +\newrgbcolor{cremark}{0.95 0.95 0.9} +\newrgbcolor{cproof}{0.95 0.92 0.95} % roman numbers \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}} @@ -94,25 +106,30 @@ % vectors are simply represented by bold font characters \renewcommand{\vec}[1]{{\bf #1{}}} - -\newtheorem{theorem}{Theorem}[section] -\newtheorem{lemma}[theorem]{Lemma} -\newtheorem{proposition}[theorem]{Proposition} -\newtheorem{corollary}[theorem]{Corollary} - -\newenvironment{proof}[1][Proof]{\begin{trivlist} -\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}} -\newenvironment{definition}[1][Definition]{\begin{trivlist} -\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}} -\newenvironment{example}[1][Example]{\begin{trivlist} -\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}} -\newenvironment{remark}[1][Remark]{\begin{trivlist} -\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}} - -\newcommand{\qed}{\nobreak \ifvmode \relax \else -\ifdim\lastskip<1.5em \hskip-\lastskip -\hskip1.5em plus0em minus0.5em \fi \nobreak -\vrule height0.75em width0.5em depth0.25em\fi} +% +% theorem environment +% +% common +\theoremstyle{plain} +\theoremseparator{:} +\theorembodyfont{\normalfont} +% theorem +\theoremsymbol{\ensuremath{\diamondsuit}} +\shadecolor{ctheorem} +\newshadedtheorem{theorem}{Theorem}[section] +% definition +\theoremsymbol{\ensuremath{\clubsuit}} +\shadecolor{cdefinition} +\newshadedtheorem{definition}[theorem]{Definition} +% remark +\theoremsymbol{\ensuremath{\diamondsuit}} +\shadecolor{cremark} +\newshadedtheorem{remark}[theorem]{Remark} +% proof +\theoremstyle{nonumberplain} +\theoremsymbol{\rule{1ex}{1ex}} +\shadecolor{cproof} +\newshadedtheorem{proof}{Proof} % author & title \author{Frank Zirkelbach} @@ -126,7 +143,6 @@ \frontmatter{} \include{title} \include{contents} -%\include{lists} \mainmatter{} \include{intro} @@ -142,6 +158,7 @@ \include{math_app} \backmatter{} +\include{lists} \include{literature} \include{ack} diff --git a/physics_compact/qm.tex b/physics_compact/qm.tex index 8e65f4c..bf8b2ee 100644 --- a/physics_compact/qm.tex +++ b/physics_compact/qm.tex @@ -6,13 +6,18 @@ \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. + +\begin{theorem}[Variational method] 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 +\end{theorem} + +\begin{proof} +The trial function $|\tilde 0\rangle$ 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{ ,} @@ -38,6 +43,8 @@ Since $\langle k|k'\rangle=\delta_{k k'}$, $H|k\rangle = E_k|k\rangle$ and $E_k\ \label{sec:vm_f} \end{equation} which proofs the variational theorem. +\end{proof} + 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} diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index 30e078e..ed9bddb 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -64,7 +64,7 @@ n_0(\vec{r})=\int \Psi_0^*(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N) \end{equation} In 1964, Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}. -{\begin{theorem} +\begin{theorem}[Hohenberg / Kohn] For a nondegenerate ground state, the ground-state charge density uniquely determines the external potential in which the electrons reside. \end{theorem} @@ -102,6 +102,6 @@ E_1 + E_2 < E_2 + E_1 + \int n(\vec{r}) \left( V_2(\vec{r})-V_1(\vec{r}) \right) d\vec{r} }_{=0} \end{equation} -is revealed, which proofs the Hohenberg Kohn theorem. \qed +is revealed, which proofs the Hohenberg Kohn theorem.% \qed \end{proof} diff --git a/physics_compact/title.tex b/physics_compact/title.tex index 523e6fd..f3162e6 100644 --- a/physics_compact/title.tex +++ b/physics_compact/title.tex @@ -26,7 +26,8 @@ \vspace{60pt} {\Large - Augsburg, \today + Augsburg\\ + \today } \end{center}