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\title{MA3441 Problem Set 2}
\author{Mark Allen}
\date{Issued: 10th October 2012\\ Due: 17th October 2012} % Activate to display a given date or no date (if empty),
         % otherwise the current date is printed 

\begin{document}
\maketitle

\section*{Disclaimer}

Everything I've written might be completely wrong. Check that what I wrote matches with what Prof. Fry wrote, especially if you keep getting the wrong answer.

\section*{Problems}
\begin{enumerate}
\item
The observable Q has the complete set of discrete normalizable eigenstates $\ket{n}$ with eigenvalues $q_n$:
\[
Q\ket{n}  = q_n\ket{n}, \braket{n}{m} = \delta_{mn}
\]
The completeness relation for this basis is
\[
 \sum\limits_{k}\ket{n}\bra{n} =  1 \text{ ( $\equiv$ unit operator).}
\]
Provide a derivation for each of the following inequalities:
\begin{enumerate}
\item
For any observable $P$, the expectation value of $P^2$ in the state $\ket{n}$, $\bra{n}P^2\ket{n}$, satisfies
\[
\bra{n}P^2\ket{n} \geq |\bra{n}P\ket{m}|^2
\]
where $m$ is any state from the above basis.
\item
For any normalisable state vector $\ket{\psi}$,
\[
\bra{\psi}Q\ket{\psi} \geq q_0 \braket{\psi}{\psi}
\]
where $q_0$ is the smallest of the eigenvalues $q_n$ belonging to the eigenstates $\ket{n}$ of $Q$.
\item
For any two normalised state vectors $\ket{\psi}$ and $\ket{\varphi}$, the real pat of their inner product, $\braket{\varphi}{\psi}$ satisfies
\[
-1 \leq \Re\braket{\varphi}{\psi} \leq 1
\]
\item
For any observable $P$ and normalised state vector $\ket{\psi}$,
\[
\bra{\psi}P^2\ket{\psi} \geq \bra{\psi}P\ket{\psi}^2
\]
\end{enumerate}
\item
The Hamiltonian operator for a two-state system is given by
\[
H = a(\ket{1}\bra{1} - \ket{2}\bra{2} + \ket{1}\bra{2} + \ket{2}\bra{1})
\]
where $a$ is a real number with the dimension of energy. Find the energy eigenvalues and the corresponding energy eigenstates.

Ans:
\[
\ket{\sqrt{2}a} = \frac{1}{2} (2 + \sqrt{2})^{1/2} \ket{1} + \frac{1}{2} (2 + \sqrt{2})^{1/2}(\sqrt{2} - 1) \ket{2}
\]
\[
\ket{-\sqrt{2}a} = \frac{1}{2} (2 + \sqrt{2})^{1/2}(1 - \sqrt{2}) \ket{1} + \frac{1}{2} (2 + \sqrt{2})^{1/2} \ket{2}
\]
\item
If $A$,$B$ = Hermitian, show that $\bra{\varphi}AB\ket{\psi}^* = \bra{\psi}BA\ket{\varphi}$
\item
Given the non-Hermitian matrix $ \left( \begin{smallmatrix} 0&9 \\1 &0 \end{smallmatrix} \right)$, Investigate
\begin{enumerate}
\item
its eigenvalues;
\item
its eigenvectors, i.e., find them;
\item
their orthogonality;
\item
and their completeness.
\end{enumerate}
\end{enumerate}
\end{document}
