= Two-level quantum return probability
A normalized <quantum state> $\chi=\sqrt q\,\psi_1+e^{i\eta}\sqrt{1-q}\,\psi_2$ in two orthonormal <energy eigenstates> returns to its original one-dimensional measurement subspace with probability
$$
|\langle\chi,e^{-iHt/\hbar}\chi\rangle|^2=q^2+(1-q)^2+2q(1-q)\cos((E_1-E_2)t/\hbar).
$$
This follows by applying <unitary time evolution> to each <energy eigenstate>, then the <Born rule>. The relative phase $\eta$ cancels from the return amplitude. Degenerate energies make the probability identically one.
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