- #1
Pietjuh
- 76
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I have to make a homework problem about neutrino oscillations, but I already don't know how to answer the first question.
Let [tex]\Psi_i[/tex], i = 1,2 be two spinor fields, with field equation
[tex]\gamma^{\mu}\partial_{\mu}\Psi_i = - \sum_{j=1}^2 M_{ij} \Psi_j[/tex]
where [tex]M_{ij}[/tex] is a hermitian matrix. Suppose this matrix has eigenvalues m1 and m2. Show that the eigenspinors of this matrix represent particles with mass [tex]m_1 = \frac{\hbar\mu_1}{c}[/tex] and [tex]m_2 = \frac{\hbar\mu_2}{c}[/tex]
I know that each of the two spinorfields have to satisfy the dirac equation. But the field equations of these spinorfields are coupled equations, so I can't just make the correspondence. From the dirac equation I know that:
[tex]i \gamma^{\mu}\partial_{\mu}\Psi_i - \frac{m_i c}{\hbar}\Psi_i = 0[/tex]
This gives me that the field equations equal to:
[tex]
\sum_{j=1}^2 M_{ij}\Psi_j = - \frac{m_i c}{i\hbar}\Psi_i
[/tex]
But if we now compose a new vector, consisting of the two spinorfields joined in one, we get the following:
[tex]
M \Psi = - \frac{c}{i\hbar} \left(\begin{array}{cc} m_1 e_4 & 0 \\
0 & m_2 e_4\end{array}\right)
[/tex]
But I don't really know how we can determine from this the masses of the particles. Can someone give me a hint?
Let [tex]\Psi_i[/tex], i = 1,2 be two spinor fields, with field equation
[tex]\gamma^{\mu}\partial_{\mu}\Psi_i = - \sum_{j=1}^2 M_{ij} \Psi_j[/tex]
where [tex]M_{ij}[/tex] is a hermitian matrix. Suppose this matrix has eigenvalues m1 and m2. Show that the eigenspinors of this matrix represent particles with mass [tex]m_1 = \frac{\hbar\mu_1}{c}[/tex] and [tex]m_2 = \frac{\hbar\mu_2}{c}[/tex]
I know that each of the two spinorfields have to satisfy the dirac equation. But the field equations of these spinorfields are coupled equations, so I can't just make the correspondence. From the dirac equation I know that:
[tex]i \gamma^{\mu}\partial_{\mu}\Psi_i - \frac{m_i c}{\hbar}\Psi_i = 0[/tex]
This gives me that the field equations equal to:
[tex]
\sum_{j=1}^2 M_{ij}\Psi_j = - \frac{m_i c}{i\hbar}\Psi_i
[/tex]
But if we now compose a new vector, consisting of the two spinorfields joined in one, we get the following:
[tex]
M \Psi = - \frac{c}{i\hbar} \left(\begin{array}{cc} m_1 e_4 & 0 \\
0 & m_2 e_4\end{array}\right)
[/tex]
But I don't really know how we can determine from this the masses of the particles. Can someone give me a hint?