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 probabilityThis 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.