From: hackbard <hackbard@hackdaworld.org>
Date: Wed, 15 Feb 2012 13:30:32 +0000 (+0100)
Subject: theorems and hyperrefs
X-Git-Url: https://hackdaworld.org/gitweb/?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}