theorems and hyperrefs

[lectures/latex.git] / physics_compact / qm.tex

diff --git a/physics_compact/qm.tex b/physics_compact/qm.tex
index 8e65f4c..bf8b2ee 100644 (file)
--- a/physics_compact/qm.tex
+++ b/physics_compact/qm.tex
@@ -6,13 +6,18 @@
 \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.
+
+\begin{theorem}[Variational method]
 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
+\end{theorem}
+
+\begin{proof}
+The trial function $|\tilde 0\rangle$ 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{ ,}
@@ -38,6 +43,8 @@ Since $\langle k|k'\rangle=\delta_{k k'}$, $H|k\rangle = E_k|k\rangle$ and $E_k\
 \label{sec:vm_f}
 \end{equation}
 which proofs the variational theorem.
+\end{proof}
+
 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}