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In the system of up-spin and down-spin states of an electron, we can write a general state of electron at time $t$ as

$$\left|\psi(t)\right>=a\left|\uparrow\right>+b\left|\downarrow\right>,$$

where $\left|\uparrow\right>$ is up-spin state and $\left|\downarrow\right>$ is down-spin state. We also say that $e^{-}$ can oscillate between these two spin states (correct me if I am wrong). Now coming to neutrino oscillation phenomenon, we can write a general state of the neutrino as

$$\left|\Psi(t)\right>=a\left|\nu_e(0)\right>+b\left|\nu_{\mu}(0)\right>.$$

Now, my question is, why do we need mass eigenstates ($\nu_1$ and $\nu_2$) to describe neutrino oscillations, while we don't need such type of thing to explain the oscillation (again I would say, correct me if I am wrong) between $e^{-}$'s up-spin state and down-spin state?

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1) the spin states for the electron are two possible eigen values +1/2 or -1/2 .

2) the neutrino states involve two masses for the neutrino

Spin is a conserved angular momentum variable.

Mass is not a conserved quantity.

A free particle carries a number of conserved quantum numbers and momentum and energy and angular momentum are also conserved. When leaving an interaction vertex , from angular momentum conservation it will be in one of the two spin states, and cannot change it until it interacts again. The state you have written describes an electron only if angular momentum is not an eigenstate of the interacting system under consideration.

Because mass is not a conserved quantity but only the total energy momentum vector leaving the interaction vertex, and the masses of neutrinos are not associated with charges ( charge is a conserved quantum number) oscillations can happen.( It is related mathematically to how virtual particles have to conserve quantum numbers but the exchanged particle is off mass shell).

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  • $\begingroup$ can you please explain one of your statements viz, "The state you have written describes an electron only if angular momentum is not an eigenstate of the interacting system under consideration." $\endgroup$ – user176263 Feb 25 '18 at 5:39
  • $\begingroup$ I mean that we do not know the interaction from which it came., whence the spin would be unique. It is a free non interacting electron, which when measured will give one of the two spin states, because spin is a quantum number. It is not the spin that is arbitrary. Mass is not a quantum number. There is no format by which an up spin can become a down spin without and interaction. Have read en.wikipedia.org/wiki/Neutral_particle_oscillation $\endgroup$ – anna v Feb 25 '18 at 6:19

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