% Homework for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2011/9/27

\documentclass[a4paper,12pt]{article}

\usepackage{pm1}

\setcounter{homework}{6}

\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{VectorSpace}
Consider the set of functions satisfying the boundary condition
\begin{equation}
\fn{\phi}{a} = \fn{\phi}{b} = \phi_0
\end{equation}
Does it form a vector space?

\question{HermitianConjugate}
Show that
\begin{enumerate}
\item
\begin{equation}
A^{\dagger\dagger} = A
\end{equation}
\item
\begin{equation}
(AB)^\dagger = B^\dagger A^\dagger
\end{equation}
\item
\begin{equation}
\left( A^\dagger \right)^{-1} = \left( A^{-1} \right)^\dagger
\end{equation}
\end{enumerate}

\question{operators}
Show that
\begin{enumerate}
\item
$A A^\dagger$ is Hermitian,
\item
$e^{iH}$ is unitary,
\item
$U_1 U_2$ is unitary.
\end{enumerate}
where $A$ is a linear operator, $H$ is a Hermitian operator, and $U_1$ and $U_2$ are unitary operators.

\question{phipsi}
Describe the properties of
\begin{equation}
\frac{\ket\phi \bra\psi}{\braket{\psi}{\phi}}
\end{equation}

\end{homework}

\end{document}

