= Solution
An operator $Q$ is Hermitian when
$$
\langle\phi,Q\psi\rangle=\langle Q\phi,\psi\rangle
$$
on its domain. Hermitian operators represent quantum observables because their expectation values and eigenvalues are real. In a normalized state,
$$
(\Delta Q)_\psi
=\left\langle\bigl(Q-\langle Q\rangle_\psi\bigr)^2
\right\rangle_\psi^{1/2}.
$$
Positivity of the norm of $(Q+i\lambda P)\psi$ for every real $\lambda$ gives
$$
0\leq
\langle Q^2\rangle_\psi
+\lambda\langle i[Q,P]\rangle_\psi
+\lambda^2\langle P^2\rangle_\psi.
$$
The middle expectation is real because $i[Q,P]$ is Hermitian. The discriminant of this quadratic is nonpositive, so
$$
\langle Q^2\rangle_\psi\langle P^2\rangle_\psi
\geq\frac14|\langle i[Q,P]\rangle_\psi|^2.
$$
Apply this to the centered operators $Q-\langle Q\rangle_\psi$ and $P-\langle P\rangle_\psi$, whose <commutator> is still $[Q,P]$. This yields the <Robertson uncertainty principle>
$$
\boxed{(\Delta Q)_\psi(\Delta P)_\psi
\geq\frac12|\langle i[Q,P]\rangle_\psi|}.
$$
Noncommuting observables therefore cannot both have arbitrarily sharply concentrated measurement distributions in the same state.
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