= Solution
Each <energy eigenstate> picks up its own phase under <unitary time evolution>. The relative phase therefore changes at $(E_1-E_0)/\hbar=\omega$. After removing an irrelevant <global phase>, the state is
$$
|\Psi(t)\rangle=\cos\theta|0\rangle+
e^{i(\phi-\omega t)}\sin\theta|1\rangle.
$$
The diagonal position <matrix elements> vanish, but the <annihilation operator> and <creation operator> give
$$
\langle0|x|1\rangle=\langle1|x|0\rangle
=\sqrt{\frac{\hbar}{2m\omega}}.
$$
The two off-diagonal cross terms in the <expectation value> yield
$$
\boxed{\langle x\rangle_t=
\sqrt{\frac{\hbar}{2m\omega}}\sin(2\theta)\cos(\phi-\omega t).}
$$
Thus the <angular frequency> is $\omega$, and the magnitude of the oscillation, meaning its amplitude, is
$$
\boxed{A=\sqrt{\frac{\hbar}{2m\omega}}\,|\sin(2\theta)|.}
$$
The peak-to-peak displacement is $2A$. The phase $\phi$ fixes where the oscillation starts but does not change $A$. A pure $|0\rangle$ or $|1\rangle$ preparation has no mean oscillation, whereas an equal-weight <quantum superposition> maximizes its amplitude within this two-state family. The motion is produced by interference between <energy eigenstates>, rather than by a changing occupation of either one.
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