From: hackbard <hackbard@hackdaworld.org>
Date: Fri, 20 Jan 2012 13:10:46 +0000 (+0100)
Subject: var meth
X-Git-Url: https://hackdaworld.org/gitweb/?a=commitdiff_plain;h=f2dddd61b086ccbe46fa103d509903c3a426c034;p=lectures%2Flatex.git

var meth
---

diff --git a/physics_compact/app.tex b/physics_compact/app.tex
deleted file mode 100644
index ccee46e..0000000
--- a/physics_compact/app.tex
+++ /dev/null
@@ -1,8 +0,0 @@
-\part{Appendices}
-
-\chapter{Mathematical tools}
-
-\section{Spherical coordinates}
-
-\section{Fourier integrals}
-
diff --git a/physics_compact/math.tex b/physics_compact/math.tex
new file mode 100644
index 0000000..1903ab1
--- /dev/null
+++ b/physics_compact/math.tex
@@ -0,0 +1,6 @@
+\chapter{Mathematical tools}
+
+\section{Spherical coordinates}
+
+\section{Fourier integrals}
+
diff --git a/physics_compact/phys_comp.tex b/physics_compact/phys_comp.tex
index a7b42a5..3790e0f 100644
--- a/physics_compact/phys_comp.tex
+++ b/physics_compact/phys_comp.tex
@@ -1,5 +1,5 @@
 \pdfoutput=0
-\documentclass[twoside,a4paper,11pt]{book}
+\documentclass[twoside,a4paper,11pt,openany]{book}
 %\documentclass[twoside,a4paper,11pt,draft]{book}
 \usepackage[activate]{pdfcprot}
 \usepackage{verbatim}
@@ -113,9 +113,11 @@
 \include{qm}
 \include{stat}
 \include{solid}
+\include{sim}
 
 \appendix{}
-\include{app}
+\part{Appendices}
+\include{math}
 
 \backmatter{}
 \include{literature}
diff --git a/physics_compact/qm.tex b/physics_compact/qm.tex
index b33a700..8e65f4c 100644
--- a/physics_compact/qm.tex
+++ b/physics_compact/qm.tex
@@ -5,5 +5,40 @@
 \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}
 
diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 50df24d..6c25ff1 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -18,7 +18,7 @@
 
 \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}.
@@ -55,5 +55,5 @@ 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.
+is revealed, which proofs the Hohenberg Kohn theorem.