From: hackbard 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.