= Solution
{c}
In the ordered <orthonormal basis> $(\psi_1,\psi_2)$, the <Hamiltonian operator> has <matrix>
$$
H=\hbar\omega\begin{pmatrix}0&1\\1&0\end{pmatrix}=\hbar\omega\sigma_1,
$$
where $\sigma_1$ is a <Pauli matrix>. Its symmetric and antisymmetric <eigenvectors> give
$$
\boxed{E_+=+\hbar\omega,\qquad \chi_+=\frac{\psi_1+\psi_2}{\sqrt2};\qquad
E_-=-\hbar\omega,\qquad \chi_-=\frac{\psi_1-\psi_2}{\sqrt2}.}
$$
The <energy eigenstates> $\chi_\pm$ are normalized and <orthogonal>; each <energy eigenvalue> is nondegenerate. Their full <stationary states> are $e^{-i\omega t}\chi_+$ and $e^{i\omega t}\chi_-$. Since they form an <orthonormal basis>, the general solution to the <Time-dependent Schrödinger equation> is
$$
\boxed{\Psi(t)=c_+e^{-i\omega t}\chi_++c_-e^{i\omega t}\chi_-,\qquad c_+,c_-\in\mathbb C.}
$$
For a normalized <quantum state>, $|c_+|^2+|c_-|^2=1$; the constants are determined by its initial <inner products> with $\chi_\pm$.
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