= Solution
Here $H_2=\begin{pmatrix}0&i\\-i&0\end{pmatrix}$ and $H_2^2=I$. Its time evolution for an additional interval $\tau$ is
$$
U_2(\tau)=e^{-iH_2\tau/\hbar}
=I\cos(\tau/\hbar)-iH_2\sin(\tau/\hbar).
$$
At $\tau=T=\pi\hbar/2$, $U_2(T)=-iH_2$. Acting on the state from the previous part gives
$$
\boxed{\Psi(2T)=(-iH_2)(-i\chi_1)=-i\chi_0}.
$$
It differs from the initial state only by a <global phase>. For every observable or projection the phase cancels between bra and ket, so \b[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.
Back to article page
