= Solution
A <pure state> is a unit vector in a complex <Hilbert space>, with vectors differing by a <global phase> representing the same physical state. An <observable> is a <self-adjoint operator>, with <measurement> probabilities determined by its <spectral projections>. For vectors in the relevant <operator domains>,
$$
\boxed{E_\psi[A]=\langle\psi,A\psi\rangle,\qquad
\Delta_\psi[A]^2=\langle\psi,A^2\psi\rangle-E_\psi[A]^2
=\|(A-E_\psi[A])\psi\|^2.}
$$
A squared norm vanishes exactly when its vector vanishes. Therefore zero <dispersion> is equivalent to $A\psi=E_\psi[A]\psi$.
In the position representation, $(X\psi)(x)=x\psi(x)$ and $(P\psi)(x)=-i\hbar\psi'(x)$. On a common differentiable invariant domain,
$$
(XP-PX)\psi=-i\hbar x\psi'+i\hbar(x\psi)'=i\hbar\psi,
$$
so $\boxed{[X,P]=i\hbar I}$.
For two <observables> set $a=\|A\psi\|^2$, $b=\|B\psi\|^2$, $c=E_\psi[i[A,B]]$. For real $t$,
$$
0\leq\|(A-itB)\psi\|^2=a-ct+bt^2.
$$
If $b>0$, nonpositivity of its discriminant yields $\boxed{4ab\geq c^2}$. Equality holds exactly when the minimum squared norm vanishes, namely
$$
\boxed{A\psi=itB\psi\quad\text{for some real }t,\quad t=c/(2b).}
$$
The degenerate case must also be included: if $B\psi=0$, then $b=c=0$ and equality holds for every $A\psi$. Thus the complete criterion is that $B\psi=0$, or that the displayed real-$t$ relation holds. This includes $A\psi=0$ by choosing $t=0$ when $b>0$.
Apply the inequality to centred operators $A=P-E_\psi[P]I$, $B=X-E_\psi[X]I$. Their squared norms are the <variances> and their <commutator> is $[P,X]=-i\hbar I$, giving
$$
\boxed{\Delta_\psi[P]^2\Delta_\psi[X]^2\geq\frac{\hbar^2}{4}.}
$$
An <eigenvector> of either operator would have its corresponding <dispersion> zero, contradicting this positive lower bound if the other <dispersion> were finite and the stated domain conditions held. In the usual $L^2(\mathbb R)$ representation there are in fact no <normalizable> exact position or momentum <eigenvectors>; their generalized <eigenvectors> are not counterexamples. The equality condition above is the <uncentered second-moment commutator inequality>.
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