 Solution

Write $\bar{\chi }=({\chi }_1+{\chi }_2)/2$ and $\delta \chi ={\chi }_2-{\chi }_1$. Since the supplied rotation matrix is real and orthogonal, $U^{-1}=U^T$. Direct matrix multiplication gives the flavor-basis HamiltonianThe common term $\bar{\chi }I$ produces only an overall quantum phase. The off-diagonal elements $H_{e\mu }^{\prime }=H_{\mu e}^{\prime }=(\delta \chi /2)\sin 2\theta$ couple the two interaction eigenstates in the Schrödinger equation. Flavor change requires both $\delta \chi \ne 0$ and $\sin 2\theta \ne 0$. For ultrarelativistic neutrinos in vacuum, $\delta \chi \simeq \Delta m^2/(2E)$, so a nonzero squared-mass difference is needed. Observable neutrino oscillation also requires coherent production and propagation of the two components and a resolvable nonzero relative phase; special propagation times can return the initial flavor even when these Hamiltonian conditions hold.

A useful distinction for the next part is that $H_{ee}^{\prime }$ is an energy, not a time-dependent survival probability amplitude. The latter is the $ee$ element of $S(t)=\exp (-i{H}^{\prime }t)$.