Probability of measuring flavor f_1 for neutrinos with different masses?

[jfy4]

Homework Statement

Neutrinos are created in states of one of two possible flavors, f_1 or f_2. Each flavor state can be expressed as a linear combination of mass eigenstates with masses m_1 and m_2
<br /> |f_1\rangle = |m_1\rangle a_{11}+|m_2\rangle a_{21}<br />
<br /> |f_2\rangle = |m_1\rangle a_{12} + |m_2\rangle a_{22}<br />
The unitary matrix a_{ij} is called the mixing matrix. The different mass of neutrinos with the same momentum move at different speeds. Eventually the light neutrino (m_1) will outrun the heavier neutrino (m_2). When the lighter neutrino reaches a detector only the flavor can be detected. what is the probability of measuring the flavor f_1? After the flavor f_1 is measured and selected, what is the probability that the neutrino continues moving with the lighter mass (m_1)?

Homework Equations

<br /> P=\frac{\langle a|b\rangle\langle b|a\rangle}{\langle a|a\rangle\langle b|b\rangle}<br />

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 |m_1\rangle addresses the detector and I want to measure the probability of it being in a state f_1. I interpret this as
<br /> \begin{align}<br /> P(f_1) &amp;= \langle m_1|f_1\rangle\langle f_1|m_1\rangle \\<br /> &amp;= (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<br /> \end{align}<br />
Then I assumed that \langle m_1|m_2\rangle=0 and \langle m_1|m_1\rangle=1 by orthogonality. Then
<br /> P(f_1)=|a_{11}|^2<br />
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.

 

[jfy4]

After the flavor f_1 is measured and selected, what is the probability that the neutrino continues moving with the lighter mass (m_1)?

Perhaps I see it differently...

Is this saying that now the state is |f_1\rangle and we want to know the probability of state |m_1\rangle, that is
<br /> \langle f_1|m_1\rangle\langle m_1|f_1\rangle ?<br />

Thanks,

 

[vela]

Yes, that's how I read it as well.

 

[jfy4]

Okay, then I took f_i=a_{ij}m_j and solved for m_j=a^{-1}_{ij}f_i. I then carried out
<br /> \langle f_1|m_1\rangle\langle m_1|f_1\rangle=\frac{|a_{22}|^2}{|a_{11}a_{22}-a_{12}a_{21}|^2}<br />
but since a_{ij} is unitary that is
<br /> =|a_{22}|^2<br />

Does that seem correct?

 

[jfy4]

ignore that dross above... I'm still stumped, they seem to be asking for the same probability. To me it sounds like: there is a neutrino of mass m_1, what is the probability of it having flavor f_1? Then: There is a neutrino of flavor f_1, what is the probability of it having mass m_1? Both of these seem to be |\langle f_1|m_1\rangle|^2...

May I have a hint?

 

[vela]

I think that's right, actually.

 

[jfy4]

You think the expressions for those probabilities are both |\langle f_1 | m_1\rangle|^2? Why would someone write a question like that... ?

 

[vela]

You'll have to ask your instructor.

 

[jfy4]

Thanks for all your help vela