From: hackbard <hackbard@hackdaworld.org>
Date: Tue, 17 Jan 2012 14:29:28 +0000 (+0100)
Subject: started dft section
X-Git-Url: https://hackdaworld.org/gitweb/?a=commitdiff_plain;h=a9f93985e52272cdecf902c4b559173a80f4b41d;p=lectures%2Flatex.git

started dft section
---

diff --git a/physics_compact/Makefile b/physics_compact/Makefile
new file mode 100644
index 0000000..03c7e27
--- /dev/null
+++ b/physics_compact/Makefile
@@ -0,0 +1,21 @@
+# Makefile
+LATEX = latex
+DVIPDF = dvipdf
+BIBTEX = bibtex
+
+SRC = phys_comp.tex
+PDF = $(SRC:%.tex=%.pdf)
+
+all: $(PDF)
+
+%.dvi: %.tex
+	$(LATEX) $<
+	$(BIBTEX) $(SRC:%.tex=%)
+	$(LATEX) $<
+	$(LATEX) $<
+
+%.pdf: %.dvi
+	$(DVIPDF) $<
+	
+clean:
+	rm -f *.log *.aux *.blg *.lof *.ps *.pdf *.toc *.bbl
diff --git a/physics_compact/ack.tex b/physics_compact/ack.tex
index f446c8c..369a074 100644
--- a/physics_compact/ack.tex
+++ b/physics_compact/ack.tex
@@ -1,4 +1,4 @@
-\chapter*{Danksagung}
+\chapter*{Acknowledgements}
 \addcontentsline{toc}{chapter}{Acknowledgements}
 
 Thanks ...
diff --git a/physics_compact/app.tex b/physics_compact/app.tex
index 5063047..ccee46e 100644
--- a/physics_compact/app.tex
+++ b/physics_compact/app.tex
@@ -1,4 +1,8 @@
-\chapter{Spherical coordinates}
+\part{Appendices}
 
-\chapter{Fourier integrals}
+\chapter{Mathematical tools}
+
+\section{Spherical coordinates}
+
+\section{Fourier integrals}
 
diff --git a/physics_compact/intro.tex b/physics_compact/intro.tex
index c70be6d..5771409 100644
--- a/physics_compact/intro.tex
+++ b/physics_compact/intro.tex
@@ -3,5 +3,5 @@
 As the title suggests, the present work constitutes an attempt to summarize mathematical models and abstractions employed in modern theoretical physics.
 Focussed on solid state theory, which, however, requires a large amount of tools, the present book tries to additionally include all prerequisites in a hopefully compact way.
 
-A final remark: This is work in progress!
+A final remark: This is work in progress and might not be very usefull for the ...
 
diff --git a/physics_compact/mech.tex b/physics_compact/mech.tex
index 52cb4c0..e67554c 100644
--- a/physics_compact/mech.tex
+++ b/physics_compact/mech.tex
@@ -1,2 +1,2 @@
-\chapter{Classical mechanics}
+\part{Classical mechanics}
 
diff --git a/physics_compact/qm.tex b/physics_compact/qm.tex
index eba2a1f..b33a700 100644
--- a/physics_compact/qm.tex
+++ b/physics_compact/qm.tex
@@ -1 +1,9 @@
-\chapter{Quantum mechanics}
+\part{Quantum mechanics}
+
+\chapter{Fundamental concepts}
+
+\section{Variational method}
+\label{sec:var_meth}
+
+\chapter{Quantum dynamics}
+
diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index f7cf91f..20ecfef 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -1 +1,47 @@
-\chapter{Theory of the solid state}
+\part{Theory of the solid state}
+
+\chapter{Atomic structure}
+
+\chapter{Electronic structure}
+
+\section{Noninteracting electrons}
+
+\subsection{Bloch's theorem}
+
+\section{Nearly free and tightly bound electrons}
+
+\subsection{Tight binding model}
+
+\section{Interacting electrons}
+
+\subsection{Density functional theory}
+
+\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 $V(\vec{r})$.
+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}.
+
+Suppose two potentials $V_1$ and $V_2$ exist, which yield the same electron density $n(\vec{r})$.
+The corresponding Hamiltonians are denoted $H_1$ and $H_2$ with the respective ground-state wavefunctions $\Psi_1$ and $\Psi_2$ and eigenvalues $E_1$ and $E_2$.
+Then, due to the variational principle (see \ref{sec:var_meth}), one can write
+\begin{equation}
+E_1=\langle \Psi_1 | H_1 | \Psi_1 \rangle < \langle \Psi_2 | H_1 | \Psi_2 \rangle
+\end{equation}
+Expressing $H_1$ by $H_2+H_1-H_2$
+\begin{equation}
+\langle \Psi_2 | H_1 | \Psi_2 \rangle =
+\langle \Psi_2 | H_2 | \Psi_2 \rangle +
+\langle \Psi_2 | H_1 -H_2 | \Psi_2 \rangle
+\end{equation}
+and the fact that the two Hamiltonians, which describe the same number of electrons, differ only in the potential
+\begin{equation}
+H_1-H_2=V_1(\vec{r})-V_2(\vec{r})
+\end{equation}
+one obtains
+\begin{equation}
+E_1<E2+\int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r}
+\end{equation}
+By switching the indices ...
+
diff --git a/physics_compact/stat.tex b/physics_compact/stat.tex
index 3ff1bdb..368fabd 100644
--- a/physics_compact/stat.tex
+++ b/physics_compact/stat.tex
@@ -1 +1 @@
-\chapter{Statistical mechanics}
+\part{Statistical mechanics}