 Equality case of the Heisenberg uncertainty relation

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.