Transition probability for a rotated two-level observable ($P(t)={\sin }^2(2\alpha ){\sin }^2((E_2-E_1)t/(2\hslash ))$)

Let orthonormal energy eigenvectors be ${\Psi }_1,{\Psi }_2$, and let a measured observable have eigenvectors $v_{+}=\cos \alpha {\Psi }_1+\sin \alpha {\Psi }_2$ and $v_{-}=-\sin \alpha {\Psi }_1+\cos \alpha {\Psi }_2$. If the initial quantum state is $v_{-}$, unitary time evolution multiplies each energy component by $e^{-it{E}_j/\hslash }$. Its amplitude in $v_{+}$ is $\sin \alpha \cos \alpha (e^{-it{E}_2/\hslash }-e^{-it{E}_1/\hslash })$. The Born rule gives the displayed probability. The energy gap fixes the oscillation frequency, while the angle between the measurement and energy bases fixes its amplitude. Energy degeneracy or coinciding bases makes the transition probability vanish.