= Two-level observable transition probability
{title2=$P_{-\to+}(t)=\sin^2((E_2-E_1)t/(2\hbar))$}
For two orthonormal <energy eigenstates> $\psi_1,\psi_2$, let an <observable> have distinct nondegenerate <eigenstates> $\phi_\pm=(\psi_1\pm\psi_2)/\sqrt2$. Starting in $\phi_-$, <unitary time evolution> gives $(e^{-iE_1t/\hbar}\psi_1-e^{-iE_2t/\hbar}\psi_2)/\sqrt2$. Its <probability amplitude> along $\phi_+$ is $(e^{-iE_1t/\hbar}-e^{-iE_2t/\hbar})/2$. The <Born rule> therefore gives the stated transition probability. The energy difference determines the oscillation frequency, while a common energy shift has no effect.
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