= Minimum-energy normalized oscillator mode
{title2=$q(t)=e^{-i\omega t+i\alpha}/\sqrt{2\omega}$}
For a <quantum harmonic oscillator> with $\omega>0$, a complex mode satisfying the <Wronskian normalization> $q\dot q^*-\dot q q^*=i$ can be written $q=re^{is}$ with $\dot s=-1/(2r^2)$. Its <vacuum energy> is $\dot r^2/2+1/(8r^2)+\omega^2r^2/2$. This is at least $\omega/2$, with equality exactly when $\dot r=0$ and $r^2=1/(2\omega)$. The same argument fixes the subhorizon positive-frequency normalization in the <Bunch-Davies vacuum> of a canonical cosmological mode.
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