= Solution
The two given <energy eigenstates> are orthonormal. Resolve the initial state as $\psi(0)=\cos\theta\,\psi_0-\sin\theta\,\psi_1$. The <Schrödinger equation> therefore gives
$$
\psi(t)=\cos\theta\,e^{-iE_0t/\hbar}\psi_0-\sin\theta\,e^{-iE_1t/\hbar}\psi_1.
$$
The second component, which is the <probability amplitude> for the requested final state, is $\sin\theta\cos\theta(e^{-iE_0t/\hbar}-e^{-iE_1t/\hbar})$. By the <Born rule>,
$$
\boxed{P(t)=\sin^2(2\theta)\sin^2\!\left(\frac{(E_1-E_0)t}{2\hbar}\right).}
$$
This includes $P(t)=0$ when the <energy eigenvalues> coincide or when the initial state is itself an <energy eigenstate>.
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