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## Homework Statement

Consider solar neutrinos of energy 1 MeV (

**EDIT: 10 MeV not 1 MeV**) which are formed at the center of the sun in the ##\nu_2## eigenstate. What fraction of it do you expect to arrive at earth as ##\nu_\mu## and what fraction as ##\nu_\tau##? Assume that it evolves adiabaticaly inside the sun.

## Homework Equations

$$

\left| \psi(x, t) \right>

= \left| \nu_2 \right> e^{-i\phi_2}$$

## The Attempt at a Solution

Because the neutrino is formed in the ##\left| \nu_2 \right>## eigenstate,

$$

\left| \psi(x, t) \right>

= \left| \nu_2 \right> e^{-i\phi_2} \\

= \left( U_{e_2}^* \left| \nu_e \right> + U_{\mu_2}^* \left| \nu_\mu \right> + U_{\tau_2}^* \left| \nu_\tau \right>\right) e^{-i\phi_2} \\

= \left( c_e \left| \nu_e \right> + c_\mu \left| \nu_\mu \right> + c_\tau \left| \nu_\tau \right>\right) e^{-i\phi_2}

$$

Then,

$$

P(\nu_e \to \nu_\mu )

= \left| \left< \nu_\mu | \psi(x, t) \right> \right|^2 \\

= \left| c_\mu \right|^2 \\

= \left| U_{\mu_2}^* \right|^2

$$

But this probability depends neither on the neutrino energy nor on the length it has traveled. What did I do wrong?

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