(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Neutrinos are created in states of one of two possible flavors, [itex]f_1[/itex] or [itex]f_2[/itex]. Each flavor state can be expressed as a linear combination of mass eigenstates with masses [itex]m_1[/itex] and [itex]m_2[/itex]

[tex]

|f_1\rangle = |m_1\rangle a_{11}+|m_2\rangle a_{21}

[/tex]

[tex]

|f_2\rangle = |m_1\rangle a_{12} + |m_2\rangle a_{22}

[/tex]

The unitary matrix [itex]a_{ij}[/itex] is called the mixing matrix. The different mass of neutrinos with the same momentum move at different speeds. Eventually the light neutrino ([itex]m_1[/itex]) will outrun the heavier neutrino ([itex]m_2[/itex]). When the lighter neutrino reaches a detector only the flavor can be detected. what is the probability of measuring the flavor [itex]f_1[/itex]? After the flavor [itex]f_1[/itex] is measured and selected, what is the probability that the neutrino continues moving with the lighter mass ([itex]m_1[/itex])?

2. Relevant equations

[tex]

P=\frac{\langle a|b\rangle\langle b|a\rangle}{\langle a|a\rangle\langle b|b\rangle}

[/tex]

3. The attempt at a solution

I feel like the two questions are the same... It sounds like to me that a neutrino prepared in a state [itex]|m_1\rangle[/itex] addresses the detector and I want to measure the probability of it being in a state [itex]f_1[/itex]. I interpret this as

[tex]

\begin{align}

P(f_1) &= \langle m_1|f_1\rangle\langle f_1|m_1\rangle \\

&= (a_{11}\langle m_1|m_1\rangle + a_{21}\langle m_1|m_2\rangle)(a_{11}^{\ast}\langle m_1|m_1\rangle + a_{21}^{\ast}\langle m_2|m_1\rangle

\end{align}

[/tex]

Then I assumed that [itex]\langle m_1|m_2\rangle=0[/itex] and [itex]\langle m_1|m_1\rangle=1[/itex] by orthogonality. Then

[tex]

P(f_1)=|a_{11}|^2

[/tex]

The problem is that this seems to me to be the way to answer both questions... Where is my misunderstanding?

Thanks,

PS I know it says this already but please don't tell me the answer, I really want to figure this out on my own, thanks.

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# Measurements of neutrinos

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