Neutrino Oscillation Survival Probability

In summary, the conversation discusses deriving the probability of a neutrino species surviving an oscillation. The equations and attempt at a solution are shown, but there is a missing step in the solution. The solution is eventually shown to be 1-\sin^{2}2\theta\sin^{2}\left(\frac{\Delta m^{2}t}{4p}\right) after filling in the blank with \frac{2\sin^{2}2\theta\cos\frac{\Delta m^{2}t}{2p}}{4}+\frac{3+\cos4\theta}{4}.
  • #1
Dahaka14
73
0

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 :smile:.
 
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  • #2
Dahaka14 said:
[tex]
&=\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
[/tex]

Shouldn't

[tex]\frac{1}{2}\sin^{2}2\theta[/tex]

be

[tex]\left( \frac{1}{2}\sin2\theta \right)^2 ?[/tex]

Then, I think it works.
 
  • #3
Okay, so now I have
[tex]\begin{align*}
&=\frac{1}{2}\sin^{2}2\theta\cos\frac{\Delta m^{2}t}{2p}+\cos^{4}\theta+\sin^{4}\theta \\
&=\frac{2\sin^{2}2\theta\cos\frac{\Delta m^{2}t}{2p}}{4}+\frac{3+\cos4\theta}{4}
\end{align*}[/tex]

I'm sorry if it might be obvious, but I can't see it. I've just looked at it for too long.
 
  • #4
There could be more than one way to show this. Here's one way: what does [itex]\left( \cos^2 \theta + \sin^2 \theta \right)^2[/itex] equal?
 
  • #5
Thanks a lot!
 

What is neutrino oscillation survival probability?

Neutrino oscillation survival probability is a measure of the probability that a neutrino will maintain its original flavor (electron, muon, or tau) as it travels through space. Neutrinos are known to change flavors as they travel, and the survival probability describes the likelihood that a neutrino will remain in its original flavor state.

How is neutrino oscillation survival probability calculated?

The calculation of neutrino oscillation survival probability involves complex mathematical equations and requires an understanding of quantum mechanics and particle physics. In general, it takes into account factors such as the distance traveled by the neutrino, the energy of the neutrino, and the masses of the different neutrino flavors.

What is the significance of neutrino oscillation survival probability?

Neutrino oscillation survival probability is significant because it provides evidence for the existence of neutrino mass and flavor oscillations. These phenomena were first theorized in the 1960s, but it wasn't until the late 20th century that experimental evidence was found to support the theory. The study of neutrino oscillations has greatly expanded our understanding of the fundamental properties of these elusive particles.

What are the implications of neutrino oscillation survival probability?

The implications of neutrino oscillation survival probability are far-reaching and have implications for many areas of physics. Understanding neutrino oscillations and their survival probability can help us better understand the properties of neutrinos, the nature of matter and antimatter, and the evolution of the universe.

How is neutrino oscillation survival probability measured?

Neutrino oscillation survival probability is typically measured in large-scale experiments, such as the Super-Kamiokande and IceCube detectors. These experiments use massive tanks of water or ice, surrounded by sensors, to detect the interactions of neutrinos. By studying the patterns of neutrino interactions, scientists can determine the survival probability and further our understanding of these elusive particles.

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