Heisenberg uncertainty relation ($\Delta x\Delta p\ge \hslash /2$)

For a normalized state with the required position and momentum moments, $[x,p]=i\hslash$ and the Robertson uncertainty principle give $\Delta x\Delta p\ge \hslash /2$. In a quantum harmonic oscillator, the arithmetic-geometric mean inequality then bounds the variance contribution to energy below by $\hslash \omega /2$.