Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S3 / 3 / 6

Choose the infinite well as $0<x<a$ with real eigenfunctions ${\psi }_n=\sqrt{2/a}\sin (n\pi x/a)$. This fixes the relative phase convention for the specified initial plus state. Each of the first two states has probability $1/2$ in the left half. Their cross integral isThe evolving superposition therefore givesSince $E_n=n^2{\pi }^2{\hslash }^2/(2m{a}^2)$, the angular frequency and ordinary frequency areThe probability varies between $1/2\pm 4/(3\pi )$ and remains in $[0,1]$. Changing the sign convention of one eigenfunction while keeping the same physical initial state requires changing its superposition coefficient too; otherwise one has prepared the other initial state and reversed the initial spatial bias.