= Transition probability for a rotated two-level observable
{title2=$P(t)=\sin^2(2\alpha)\sin^2((E_2-E_1)t/(2\hbar))$}
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 $e^{-itE_j/\hbar}$. Its amplitude in $v_+$ is $\sin\alpha\cos\alpha(e^{-itE_2/\hbar}-e^{-itE_1/\hbar})$. 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.
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