Sudden oscillator frequency quench (${\alpha }^2-{\beta }^2=1$)

A jump from positive frequency $\omega$ to $\Omega$ keeps an oscillator mode and its first derivative continuous. A normalized positive-frequency incident solution has outgoing coefficients $\alpha =(\Omega +\omega )/(2\sqrt{\Omega \omega })$ and $\beta =(\Omega -\omega )/(2\sqrt{\Omega \omega })$, satisfying ${\alpha }^2-{\beta }^2=1$. This temporal matching is not a sum-one spatial-scattering probability rule.