Initial condition: start as a pure $\nu_e$#

Set $c_e(0) = 1$, $c_\mu(0) = 0$. Inverting the rotation matrix gives the initial mass-basis amplitudes:

$c_1(0) = \cos\theta, \qquad c_2(0) = -\sin\theta$

Substituting:

$\begin{pmatrix}c_e(t) \\ c_\mu(t)\end{pmatrix} = \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}\begin{pmatrix}\cos\theta\; e^{-iE_1 t/\hbar} \\ -\sin\theta\; e^{-iE_2 t/\hbar}\end{pmatrix}$

This gives:

$c_e(t) = \cos^2\theta\; e^{-iE_1 t/\hbar} + \sin^2\theta\; e^{-iE_2 t/\hbar}$

$c_\mu(t) = \sin\theta\cos\theta\left(e^{-iE_1 t/\hbar} - e^{-iE_2 t/\hbar}\right)$

Relativistic energy approximation#

Neutrinos are ultra-relativistic: $E_i \gg m_i c^2$. For fixed momentum $p$:

$E_i = \sqrt{p^2c^2 + m_i^2c^4} \approx pc\left(1 + \frac{m_i^2 c^2}{2p^2}\right)$

So:

$E_2 - E_1 \approx \frac{(m_2^2 - m_1^2)c^4}{2E} = \frac{\Delta m^2 c^4}{2E}$

where $E \approx pc$ is the common energy. With $L = ct$ (distance traveled):

$\frac{(E_2 - E_1)t}{2\hbar} = \frac{\Delta m^2 c^4\, L}{4E\hbar c}$

Putting it all together:

$\boxed{P(\nu_e \to \nu_e) = 1 - \sin^2(2\theta)\sin^2\!\left(\frac{\Delta m^2 c^4\, L}{4E\hbar c}\right)}$

Why do oscillations imply neutrinos have mass?#

Look at the argument of the $\sin^2$: it contains $\Delta m^2 = m_2^2 - m_1^2$. If both neutrinos were massless, $\Delta m^2 = 0$ and the probability would be identically 1 — no oscillation. More precisely:

• Oscillations require two different phases $e^{-iE_1 t/\hbar}$ and $e^{-iE_2 t/\hbar}$ to interfere.
• For massless particles, $E_i = pc$ for all $i$, so both phases are identical and they never interfere.
• A non-trivial oscillation pattern therefore requires $m_1 \neq m_2$, and at least one must be non-zero.

Note: oscillations only constrain mass differences, not absolute masses. This is why neutrino oscillation experiments cannot tell us the absolute mass scale — only that $\Delta m^2 \neq 0$.

Three flavors: how many must be massive?#

In the three-flavor case there are three mass eigenstates $|\nu_1\rangle, |\nu_2\rangle, |\nu_3\rangle$ and three flavor eigenstates $|\nu_e\rangle, |\nu_\mu\rangle, |\nu_\tau\rangle$. Oscillations between all three flavor pairs have been observed.

If only one mass eigenstate were massive (say $m_1 \neq 0$, $m_2 = m_3 = 0$), we would have $\Delta m^2_{23} = 0$, meaning no oscillation between the states that mix with $|\nu_2\rangle$ and $|\nu_3\rangle$. Since oscillations between all flavors are observed, we need $\Delta m^2_{12} \neq 0$ and $\Delta m^2_{23} \neq 0$, which requires at least two massive eigenstates.