= Solution
{c}
The coefficient of the <momentum eigenstate> $\psi_2$ is $e^{-i\Omega t}/\sqrt6$. The <Born rule> therefore gives $\Pr(P=2\hbar)=1/6$. Conditional on this outcome, the <projective measurement> prepares the <post-measurement state> $e^{-i\Omega t}\psi_2$, equivalent to $\psi_2$ up to <global phase>.
Since $H\psi_2=(4\hbar^2/(2m))\psi_2$, the subsequent <energy> measurement has <conditional probability> one of giving the specified <energy eigenvalue>. The <probability> for the entire sequence is consequently
$$
\boxed{\Pr\left(P=2\hbar\ \text{then}\ E=\frac{4\hbar^2}{2m}\right)=\frac16\times1=\frac16.}
$$
Any intervening <unitary time evolution> under the same <Hamiltonian operator> preserves this <energy eigenstate>, up to <global phase>.
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