% Homework for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2010/11/18

\documentclass[a4paper,12pt]{article}

\usepackage{pm1}

\setcounter{homework}{7}

\begin{document}

\begin{homework}
[Answers should be submitted, as both a tex file and a pdf file, to both me and the teaching assistants. You may use this file as a template.]

\question{eigenvalues}
Show that the eigenvalues $\alpha$ of a
\begin{enumerate}
\item
Hermitian operator satisfy
\begin{equation}
\alpha^* = \alpha
\end{equation}
\item
unitary operator satisfy
\begin{equation}
\alpha^* = \alpha^{-1}
\end{equation}
\end{enumerate}

\question{eigenspaces}
Show that the eigenspaces of a
\begin{enumerate}
\item
Hermitian operator
\item
unitary operator
\end{enumerate}
are orthogonal.

\question{Heisenberg}
The position and momentum operators have the commutation relation
\begin{equation}
\left[ \hat{x} , \hat{p} \right] = i \hbar
\end{equation}
Show that they cannot have a common eigenvector.
Interpret physically.

\end{homework}

\end{document}

