= 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 <Hamiltonian>
$$
H'=\begin{pmatrix}\chi_1c^2+\chi_2s^2&(\chi_2-\chi_1)sc\\(\chi_2-\chi_1)sc&\chi_1s^2+\chi_2c^2\end{pmatrix}
=\boxed{\bar\chi I+\frac{\delta\chi}{2}\begin{pmatrix}-\cos2\theta&\sin2\theta\\\sin2\theta&\cos2\theta\end{pmatrix}.}
$$
The common term $\bar\chi I$ produces only an overall quantum phase. The off-diagonal elements $H'_{e\mu}=H'_{\mu e}=(\delta\chi/2)\sin2\theta$ couple the two <interaction eigenstates> in the <Schrödinger equation>. Flavor change requires both $\delta\chi\ne0$ and $\sin2\theta\ne0$. 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 $H'_{ee}$ is an <energy>, not a time-dependent survival <probability amplitude>. The latter is the $ee$ element of $S(t)=\exp(-iH't)$.
Back to article page
