= Equality case of the Heisenberg uncertainty relation
Equality requires the centred vectors $x'\psi$ and $p'\psi$ to be imaginary scalar multiples. Thus, for some $s>0$,
$$
(p-p_0)\psi=is(x-x_0)\psi.
$$
In the position representation this is a first-order differential equation, whose normalizable solutions are
$$
\psi(x)=C\exp\left(\frac{ip_0x}{\hbar}-\frac{s(x-x_0)^2}{2\hbar}\right).
$$
Hence the states saturating the position-momentum uncertainty relation are Gaussian wave packets, up to translation, modulation, and phase.
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