= Equality case of the Heisenberg uncertainty relation
Equality in the <Heisenberg uncertainty relation> requires the centred <vectors> $x'\psi$ and $p'\psi$ to be imaginary <scalar multiples>, where $x'=x-x_0I$, $p'=p-p_0I$ and $x_0,p_0$ are the <expectation values>. Thus, for some $s>0$,
$$
(p-p_0)\psi=is(x-x_0)\psi.
$$
In the <position> representation, using the <momentum operator> turns this into 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 <quantum states> saturating the <Heisenberg uncertainty relation> are these <Gaussian wave packets>, with arbitrary <position> and <momentum> means and <global phase>. Their width is $\sigma^2=\hbar/(2s)$. A <Gaussian wave packet> with an additional quadratic <quantum phase> need not saturate this product bound.
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