 Solution

Forthe phase of the $n$th stationary state isReflection in the midpoint givessoThe minus sign is a global quantum phase and therefore does not change the physical state. The relative even--odd phases first acquire this reflection pattern at $T$. Applying it twice givesIf $f$ vanishes on the right half of the well, then $\psi (x,T)$ vanishes on the left half. The Born rule consequently gives zero probability of finding the particle in $0\le x\le a/2$ at time $T$.