Solution
= Solution
{c}
The <parity operator> exchanges $\psi_n$ and $\psi_{-n}$. Their coefficients in the <wavefunction> are equal, so
$$
(R\Psi)(x,t)=\Psi(-x,t)=\frac{1+e^{-i\Omega t}\cos(-2x)}{\sqrt{3\pi}}=\Psi(x,t).
$$
Thus the <wavefunction> is a <parity operator> <eigenstate> with <eigenvalue> $+1$ at every time. The <orthogonal projection> onto the $-1$ <eigenspace> is $(I-R)/2$, which annihilates it. By the <Born rule>,
$$
\boxed{\Pr(R=-1)=0.}
$$