= Solution
The <Time-dependent Schrödinger equation> is $i\hbar\partial_t\Psi=H\Psi$. The normalized eigenstates of the Hermitian <Hamiltonian> $H_3$ with distinct <eigenvalues> are orthogonal, so they form an orthonormal <basis> of this two-state system. Substitute the expansion of $\Psi$ and compare the two coefficients:
$$
\boxed{i\hbar\dot a_0=-a_0,\qquad i\hbar\dot a_1=a_1}.
$$
Hence $a_0(t)=a_0(0)e^{it/\hbar}$ and $a_1(t)=a_1(0)e^{-it/\hbar}$. For the normalized test state $\phi=\alpha\chi_0+\beta\chi_1$, the <Born rule> gives
$$
\boxed{P_\phi(t)=|\langle\phi,\Psi(t)\rangle|^2
=|\bar\alpha a_0(t)+\bar\beta a_1(t)|^2}.
$$
The complex conjugates are essential; relative phases can affect this probability even though the <energy>-state populations remain constant under $H_3$.
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