From: hackbard 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}