Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 15B / b

Expand the normalized initial wavefunction in the orthonormal energy basis:The Time-dependent Schrodinger equation then gives the unitary time evolution

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 $T$.

The specified initial state is normalized. The requested energy is $E_2$, and its probability amplitude isThus the Born rule givesUnitary evolution changes each energy coefficient only by a phase, so this probability remains $1/2$ at both $T$ and $2T$. If an energy measurement returns $E_2$, the state collapses to the nondegenerate eigenstate ${\chi }_2$; a subsequent energy measurement therefore returns $E_2$ with probability