= Solution
Each <energy eigenstate> evolves by its own <quantum phase> $e^{-iE_mt/\hbar}$. The two $m=\pm2$ components have the same <energy>, while the $m=0$ <quantum phase> stays one. Thus
$$
|\Psi(t)\rangle=\sqrt{\frac23}|0\rangle+\frac{e^{-2i\hbar t/I}}{\sqrt6}(|2\rangle+|-2\rangle).
$$
In angle representation this is
$$
\boxed{\Psi(\phi,t)=\frac1{\sqrt{3\pi}}\left[1+e^{-2i\hbar t/I}\cos2\phi\right].}
$$
At $t=0$, $1+\cos2\phi=2\cos^2\phi$ recovers the initial state. The fixed coefficient magnitudes keep the <angular momentum> and <energy> probabilities constant even though the density in angle changes.
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