What is the physical meaning of coherent superposition of mass eigenstates of neutrinos? And why it is necessary for the oscillation of neutrinos? Why can't an incoherent superposition of mass eigenstates make neutrinos to oscillate?
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Waves have sinusoidal shapes.
$$u(x, t) = A(x, t)\sin(kx - \omega t + \phi)$$
$\phi$ ("phi") is a "phase." It is a constant that tells at what value the sine function has when $t=0$ and $x=0$.
If one happens to have two waves overlapping, then the $\phi_1 - \phi_2$ of the functions is the phase difference of the two waves. How much they differ at the beginning ($x=0$ and $t=0$), and this phase difference is evidently kept all the way through. This establishes/defines the coherence of the two waves, which can be superposed, i.e. added, and display wave patterns.
If the $\phi$ phases of waves to be added/superposed are random, the two waves will be incoherent by definition, and no wave patterns can appear.
In the case of the mass eigenstates of the neutrinos incoherence would mean that no wave pattern would survive to be measured by our instruments. Since one observes oscillations, it is evident that the superposition is of coherent waves.
