 Solution

In the ordered eigenbasis, $H_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, a Pauli matrix. The coefficient equations are $i\hslash {\dot{a}}_0=a_1$ and $i\hslash {\dot{a}}_1=a_0$, with $a_0(0)=1$, $a_1(0)=0$. Differentiating the first gives ${\ddot{a}}_0=-a_0/{\hslash }^2$, and its initial derivative vanishes. ThusThe population of ${\chi }_1$ is ${\sin }^2(t/\hslash )$. To be an $H_3$ eigenstate of eigenvalue $+1$, the coefficient of ${\chi }_0$ must vanish. Its first positive zero isThe global phase does not change its eigenstate or measurement probabilities.