From: hackbard <hackbard@hackdaworld.org>
Date: Sat, 4 Feb 2012 20:17:50 +0000 (+0100)
Subject: more on hk ... needs more attention!
X-Git-Url: https://hackdaworld.org/gitweb/?a=commitdiff_plain;h=5cf9c85d580fb9d18c1f0ab3b6647edcca0b0cf1;p=lectures%2Flatex.git

more on hk ... needs more attention!
---

diff --git a/physics_compact/phys_comp.tex b/physics_compact/phys_comp.tex
index 3790e0f..3d2423d 100644
--- a/physics_compact/phys_comp.tex
+++ b/physics_compact/phys_comp.tex
@@ -93,6 +93,26 @@
 % 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}
+
 % author & title
 \author{Frank Zirkelbach}
 \title{Physics compact}
diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 6c25ff1..3cb0480 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -18,11 +18,50 @@
 
 \subsubsection{Hohenberg-Kohn theorem}
 
+The Hamiltonian of a many-electron problem has the form
+\begin{equation}
+H=T+V+U\text{ ,}
+\end{equation}
+where
+\begin{eqnarray}
+T & = & \langle\Psi|\sum_{i=1}^N\frac{-\hbar^2}{2m}\nabla_i^2|\Psi\rangle\\
+  & = & \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
+        \langle \Psi | \vec{r} \rangle \langle \vec{r} |
+        \frac{-\hbar^2}{2m}\nabla_i^2
+        | \vec{r}' \rangle \langle \vec{r}' | \Psi \rangle\\
+  & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} \,
+        \nabla_i \Psi^*(\vec{r}) \nabla_i \Psi(\vec{r})
+        \text{ ,} \\
+V & = & V(\vec{r})\Psi^*(\vec{r})\Psi(\vec{r})d\vec{r} \text{ ,} \\
+U & = & \frac{1}{2}\int\frac{1}{\left|\vec{r}-\vec{r}'\right|}
+        \Psi^*(\vec{r})\Psi^*(\vec{r}')\Psi(\vec{r}')\Psi(\vec{r})
+        d\vec{r}d\vec{r}'
+\end{eqnarray}
+represent the kinetic energy, the energy due to the external potential and the energy due to the mutual Coulomb repulsion.
+
+\begin{remark}
+As can be seen from the above, two many-electron systems can only differ in the external potential and the number of electrons.
+The number of electrons is determined by the electron density.
+\begin{equation}
+N=\int n(\vec{r})d\vec{r}
+\end{equation}
+Now, if the external potential is additionally determined by the electron density, the density completely determines the many-body problem.
+\end{remark}
+
 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 $V(\vec{r})$.
+\begin{equation}
+n_0(\vec{r})=\int \Psi_0^*(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
+                  \Psi_0(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
+             d\vec{r}_2d\vec{r}_3\ldots d\vec{r}_N
+\end{equation}
 In 1964, Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}.
+
+{\begin{theorem}
 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}.
+\end{theorem}
 
+\begin{proof}
+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
@@ -55,5 +94,6 @@ 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.
+is revealed, which proofs the Hohenberg Kohn theorem. \qed
+\end{proof}