X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=50df24d29d03741e8a4efb3933ae5b98316dcbba;hp=f7cf91f0e439c33929274f0a1b6b5d148aeb8ff7;hb=9fac15a186e73cf587fb7d9dfa1c737c778cd08a;hpb=eb7c28aab8fa8bd2f0639625aeecfb9c3e2d94f8

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
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--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -1 +1,59 @@
-\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 \text{ .}
+\label{subsub:hk01}
+\end{equation}
+Expressing $H_1$ by $H_2+H_1-H_2$, the last part of \eqref{subsub:hk01} can be rewritten:
+\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}
+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}
+and, thus
+\begin{equation}
+E_1<E2+\int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r}
+\text{ .}
+\label{subsub:hk02}
+\end{equation}
+By switching the indices of \eqref{subsub:hk02} and adding the resulting equation to \eqref{subsub:hk02}, the contradiction
+\begin{equation}
+E_1 + E_2 < E_2 + E_1 +
+\underbrace{
+\int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r} +
+\int n(\vec{r}) \left( V_2(\vec{r})-V_1(\vec{r}) \right) d\vec{r}
+}_{=0}
+\end{equation}
+is revealed.
+