= Solution
In the ordered eigenbasis, $H_1=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, a <Pauli matrix>. The coefficient equations are $i\hbar\dot a_0=a_1$ and $i\hbar\dot a_1=a_0$, with $a_0(0)=1$, $a_1(0)=0$. Differentiating the first gives $\ddot a_0=-a_0/\hbar^2$, and its initial derivative vanishes. Thus
$$
\boxed{a_0(t)=\cos(t/\hbar),\qquad a_1(t)=-i\sin(t/\hbar)}.
$$
The population of $\chi_1$ is $\sin^2(t/\hbar)$. To be an $H_3$ eigenstate of <eigenvalue> $+1$, the coefficient of $\chi_0$ must vanish. Its first positive zero is
$$
\boxed{T=\frac{\pi\hbar}{2},\qquad\Psi(T)=-i\chi_1}.
$$
The <global phase> does not change its eigenstate or measurement probabilities.
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