Show that in a 2-flavor ( $v_{\mathrm{e}}$ and $v_\mu$ ) model of neutrino oscillations, if the two energies $E_1$ and $E_2$ are approximately equal and much larger than the neutrino masses, then ( $\hbar=c=1$ units)
$$
E_2-E_1 \simeq \frac{\Delta m^2}{2 E}
$$
where $E_1 \sim E_2 \equiv E$ and $\Delta m^2 \equiv m_2^2-m_1^2$. Thus, show that the probability for flavor survival and conversion as a function of distance traveled $r \sim c t$ may be expressed as
$$
\begin{gathered}
P\left(v_{\mathrm{e}} \rightarrow v_{\mathrm{e}}, r\right)=1-\sin ^2(2 \theta) \sin ^2\left(\frac{\pi r}{L}\right), \\
P\left(v_{\mathrm{e}} \rightarrow v_\mu, r\right)=\sin ^2(2 \theta) \sin ^2\left(\frac{\pi r}{L}\right),
\end{gathered}
$$
where $\theta$ is the mixing angle and the oscillation length $L \equiv 4 \pi E / \Delta m^2$ is the distance (assuming $v \sim c$ ) over which one complete flavor oscillation occurs.