\question
{Shotaro Maruyama}       
{2/3/12}          
{shotaro.maruyama@cmpt.phys.tohoku.ac.jp}  
{2-6}   
{Using the unitary matrix $U$ in Eq.\,(2.16), diagonalize the Hamiltonian in Eq.\,(2.14).}
{
The diagonalization can be done by a unitary transformation of the Hamiltonian matrix $H$ given by $U^{\dagger}HU$, where the Hamiltonian $H$ and the unitary matrix $U$ are given, respectively, by Eqs.\,(2.13) and (2.15):

\begin{eqnarray}
U^{\dagger}HU
& = &
\frac{1}{2} \left(\begin{array}{ccc} 1&1 \\ 1&-1\end{array}\right)
\left(\begin{array}{ccc}  E_{1s}&V_{0} \\ E_{1s}&V_{0}\end{array}\right)
\left(\begin{array}{ccc} 1&1 \\ 1&-1\end{array}\right)
\nonumber \\
& = &
\frac{1}{2}\left(\begin{array}{ccc}  E_{1s}+V_{0}&V_{0}+E_{1s} \\ E_{1s}-V_{0}&V_{0}-E_{1s}\end{array}\right)\left(\begin{array}{ccc} 1&1 \\ 1&-1\end{array}\right)
\nonumber \\
& = &
\frac{1}{2}\left(\begin{array}{ccc} 2(E_{1s}+V_{0})&0 \\ 0&2(E_{1s}-V_{0})\end{array}\right)
\nonumber \\
& = &
\left(\begin{array}{ccc} E_{1s}+V_{0}&0 \\ 0&E_{1s}-V_{0}\end{array}\right)
\end{eqnarray}
}
