Robertson uncertainty principle

$\Delta A\Delta B\ge \frac{1}{2}|⟨[A,B]⟩|$.

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 $\hslash \omega /2$.

For a state with zero position and momentum means, positivity of $\Vert (p-isx)\psi {\Vert }^2$ for every real $s$ givesThe discriminant is nonpositive, hence $\Delta x\Delta p\ge \hslash /2$.

Equality requires the centred vectors $x^{\prime }\psi$ and $p^{\prime }\psi$ to be imaginary scalar multiples. Thus, for some $s>0$,In the position representation this is a first-order differential equation, whose normalizable solutions areHence the states saturating the position-momentum uncertainty relation are Gaussian wave packets, up to translation, modulation, and phase.