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 $m_2$
$$|f_1\rangle = |m_1\rangle a_{11}+|m_2\rangle a_{21}$$
$$|f_2\rangle = |m_1\rangle a_{12} + |m_2\rangle a_{22}$$
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

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

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
$$\begin{align} P(f_1) &= \langle m_1|f_1\rangle\langle f_1|m_1\rangle \\ &= (a_{11}\langle m_1|m_1\rangle + a_{21}\langle m_1|m_2\rangle)(a_{11}^{\ast}\langle m_1|m_1\rangle + a_{21}^{\ast}\langle m_2|m_1\rangle \end{align}$$
Then I assumed that $\langle m_1|m_2\rangle=0$ and $\langle m_1|m_1\rangle=1$ by orthogonality. Then
$$P(f_1)=|a_{11}|^2$$
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
$$\langle f_1|m_1\rangle\langle m_1|f_1\rangle ?$$

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
$$\langle f_1|m_1\rangle\langle m_1|f_1\rangle=\frac{|a_{22}|^2}{|a_{11}a_{22}-a_{12}a_{21}|^2}$$
but since $a_{ij}$ is unitary that is
$$=|a_{22}|^2$$

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