From: hackbard Date: Tue, 17 Jan 2012 22:22:35 +0000 (+0100) Subject: more intro +hk proof X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=9fac15a186e73cf587fb7d9dfa1c737c778cd08a;p=lectures%2Flatex.git more intro +hk proof --- diff --git a/physics_compact/intro.tex b/physics_compact/intro.tex index 5771409..db3bf5c 100644 --- a/physics_compact/intro.tex +++ b/physics_compact/intro.tex @@ -3,5 +3,7 @@ As the title suggests, the present work constitutes an attempt to summarize mathematical models and abstractions employed in modern theoretical physics. Focussed on solid state theory, which, however, requires a large amount of tools, the present book tries to additionally include all prerequisites in a hopefully compact way. -A final remark: This is work in progress and might not be very usefull for the ... +A final remark: This is work in progress. +In the initial form, the manuscript will hardly contain pedagogically useful explanations but pure mathematical descriptions. +Thus, it is considered my own personal leaflet rather than a general introduction to solid state theory. diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index 20ecfef..50df24d 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -27,21 +27,33 @@ Suppose two potentials $V_1$ and $V_2$ exist, which yield the same electron dens 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 +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 $H_2+H_1-H_2$ +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} -and the fact that the two Hamiltonians, which describe the same number of electrons, differ only in the potential +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} -one obtains +and, thus \begin{equation} E_1