$$ \newcommand\Tr{\mathrm{Tr}} \newcommand{\braket}[2]{\langle #1 \mid #2 \rangle} \newcommand\I{\mathbb{I}} \newcommand{\avg}[1]{\left< #1 \right>} \newcommand{\RD}{D} \newcommand{\ri}{\mathrm{i}} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Sign}{Sign} \newcommand{\ii}{\mathrm i} \newcommand{\vv}{\mathrm v} \newcommand{\ff}{\mathrm f} \newcommand{\mm}{\mathrm m} \newcommand{\ee}{\mathrm e} \newcommand{\xx}{\mathrm x} \newcommand{\RR}{\mathrm R} \newcommand{\dd}{\mathrm d} \newcommand{\FF}{\mathrm F} \newcommand{\BB}{\mathrm B} \newcommand{\vph}{v_{\mathrm{ph}}} $$


Neutrino oscillations in vacuum and in matter with a smooth profile have been explained. The neutrino oscillation phenomenon reveals a secret of the nature of the neutrino, i.e., its flavor states are not the same as the mass eigenstates of the Hamiltonian. As a result, a neutrino produced in the pure flavor state through weak interaction will not remain in the same flavor state as it propagates, but oscillates between different flavors. The problem of neutrino oscillations in an environment with rapidly varying matter densities is significantly more difficult than that in a smooth profile. This will be discussed in the next chapter.

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