= Solution
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 is
$$
\int_0^{a/2}\psi_1\psi_2\,dx=\frac2\pi\int_0^{\pi/2}\sin u\sin2u\,du=\frac4{3\pi}.
$$
The evolving superposition therefore gives
$$
\boxed{P_{\rm left}(t)=\frac12+\frac4{3\pi}\cos\left(\frac{(E_2-E_1)t}{\hbar}\right).}
$$
Since $E_n=n^2\pi^2\hbar^2/(2ma^2)$, the angular frequency and ordinary frequency are
$$
\boxed{\omega=\frac{3\pi^2\hbar}{2ma^2},\qquad
f=\frac{\omega}{2\pi}=\frac{3h}{8ma^2}.}
$$
The <probability> varies between $1/2\pm4/(3\pi)$ 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.
\Image[/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-two-state-probabilities.png]
{title=Original two-state transition and left-half well <probabilities> over two recurrence periods}
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