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## Homework Statement

I'm lost at how to derive the probability of a neutrino species surviving an oscillation. After performing calculations, I can't seem to get it into the nice tidy form

[tex]1-\sin^{2}2\theta\sin^{2}\left(\frac{\Delta m^{2}t}{4p}\right)[/tex]

## Homework Equations

Whatev...

[tex]|\langle\nu_{e}|\psi(t)\rangle|^{2}[/tex]

[tex]E_{i}=\sqrt{p^{2}+m_{i}^{2}}\approx p+\frac{m_{i}^{2}}{2p},~\text{where}~p\gg m[/tex]

[tex]\text{and}~\Delta m^{2}=m_{2}^{2}-m_{1}^{2}[/tex]

## The Attempt at a Solution

[tex]\begin{align*}

P_{e\rightarrow\nu_{e}}=\langle\nu_{e}|\psi(t)\rangle&=\langle\nu_{e}|\nu_{e}\rangle e^{-iEt/\hbar}=\left|

\left(

\begin{array}{ccc}

\cos\theta & \sin\theta

\end{array} \right)

\left(

\begin{array}{ccc}

\cos\theta e^{-iE_{1}t/\hbar} \\

\sin\theta e^{-iE_{2}t/\hbar}

\end{array} \right)

\right|^{2} \\

&=|\cos^{2}\theta e^{-iE_{1}t/\hbar}+\sin^{2}\theta e^{-iE_{2}t/\hbar}|^{2} \\

&=|e^{-iE_{1}t/\hbar}(\cos^{2}\theta+\sin^{2}\theta e^{-(iE_{2}-E_{1})t/\hbar})|^{2} \\

&=(\cos^{2}\theta+\sin^{2}\theta e^{-i(E_{2}-E_{1})t/\hbar})(\cos^{2}\theta+\sin^{2}\theta e^{i(E_{2}-E_{1})t/\hbar}) \\

&=\frac{1}{2}\sin^{2}2\theta\left(\cos\frac{\Delta m^{2}t}{2p}-i\sin\frac{\Delta m^{2}t}{2p}+\cos\frac{\Delta m^{2}t}{2p}+i\sin\frac{\Delta m^{2}t}{2p}\right)+\cos^{4}\theta+\sin^{4}\theta \\

&=\sin^{2}2\theta\cos\frac{\Delta m^{2}t}{2p}+\cos^{4}\theta+\sin^{4}\theta \\

&=...? \\

&=1-\sin^{2}2\theta\sin^{2}\left(\frac{\Delta m^{2}t}{4p}\right)

\end{align*}[/tex]

Can someone help me fill in the blank? It would be best if I could do it on my own, so if possible just give me hints. If it is too explicit, then just tell me I guess. But as we all know, in order for me to truly own the idea, I should only be gently pushed toward the answer .