more intro +hk proof

diff --git a/physics_compact/intro.tex b/physics_compact/intro.tex
index 5771409..db3bf5c 100644 (file)
--- 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.
 
 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 (file)
--- 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}
 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}
 \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}
 \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}
 \begin{equation}
 H_1-H_2=V_1(\vec{r})-V_2(\vec{r})
 \end{equation}

-one obtains
+and, thus

 \begin{equation}
 E_1<E2+\int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r}
 \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}
 \end{equation}

-By switching the indices ...
+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.