 Solution

Here $H_2=\left(\begin{array}{cc}0& i\\ -i& 0\end{array}\right)$ and $H_2^2=I$. Its time evolution for an additional interval $\tau$ isAt $\tau =T=\pi \hslash /2$, $U_2(T)=-i{H}_2$. Acting on the state from the previous part givesIt differs from the initial state only by a global phase. For every observable or projection the phase cancels between bra and ket, so the final state is observationally indistinguishable from the original isolated state. A relative phase between two components or between coherently controlled alternatives would be observable; this overall phase of the entire state is not.