mass eigenstates have the same momenta. As discussed in  [@Beuthe], in order to have a more consistent description, one should use wave packets instead of plane waves. However, a wave packet analysis goes beyond the scope of this paper: it would make the calculations much more difficult, and the expected result should not be much different from the one here derived. Moreover, in the standard treatment of neutrino oscillations, the plane wave analysis already gives a satisfactory description, to some degree, both in vacuum and in matter. Mathematically, we will compute the following quantity: $$\label{eq:Tab}
T_{\alpha\rightarrow\beta}\equiv\left\langle \nu_{\beta};\mathbf{p}_{f},s_{f}\left|U\left(t\right)\right|\nu_{\alpha};\mathbf{p}_{i},s_{i}\right\rangle =\sum_{i,j=1}^{n}\text{U}_{\alpha j}\text{U}_{\beta i}^{*}\left\langle \nu_{i};\mathbf{p}_{f},s_{f}\left|U\left(t\right)\right|\nu_{j};\mathbf{p}_{i},s_{i}\right\rangle$$ here $U\left(t\right)$ is the time-evolution operator of Eq.  while $\left|\nu_{\alpha};\mathbf{p}_{i},s_{i}\right\rangle$ is the flavour eigenstate $\alpha$ with initial momentum $\mathbf{p}_{i}$ and spin $s_{i}$ and $\left|\nu_{\beta};\mathbf{p}_{f},s_{f}\right\rangle$ is the final state with momentum $\mathbf{p}_{f}$ and spin $s_{f}$. Since the Hamiltonian is the sum of Hamiltonians associated to different mass eigenstates ($H=\sum_{j=1}^{n}H_{j}$), it is convenient to expand the flavour eigenstates into the mass eigenstates, as we did in Eq. (\[eq:Tab\]). The form of the Hamiltonian also implies that $U(t)$ factorizes: $U\left(t\right)=\prod_{k=1}^{n}U_{k}\left(t\right)$. Here $U_{k}\left(t\right)$ is the time evolution operator related to the Fock space of the neutrino having a definite mass $m_{k}$. This is 