= Solution
Under <unitary time evolution>, each <energy eigenstate> acquires its own <dynamical phase>. Since $E_2=E_{-2}=E_\star$, the <wavefunction> is
$$
\boxed{\begin{aligned}
\Psi(t,x)&=\sqrt{\frac23}\,\phi_0(x)-\frac{e^{-iE_\star t/\hbar}}{\sqrt6}\bigl(\phi_2(x)+\phi_{-2}(x)\bigr)\\
&=\frac{1-e^{-i\Omega t}\cos(2x/R)}{\sqrt{3\pi R}},\qquad
\Omega=\frac{E_\star}{\hbar}=\frac{2\hbar}{mR^2}.
\end{aligned}}
$$
At $t=0$, the identity $1-\cos(2x/R)=2\sin^2(x/R)$ recovers the normalized initial <wavefunction>.
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