= Solution
Centre the <position operator> and <momentum operator> by their real <expectation values>:
$$
A=X-x_0I,\quad B=P-p_0I,\qquad x_0=\mathbb E_\psi(X),\quad p_0=\mathbb E_\psi(P).
$$
Scalar multiples of the <identity operator> commute with every <linear operator>, so the <canonical commutation relation> gives
$$
[A,B]=[X,P]=i\hbar I,\qquad C=\hbar I,\qquad \mathbb E_\psi(C)=\hbar.
$$
The second moments of the centred <linear operators> are the <variances>, $\mathbb E_\psi(A^2)=\Delta_\psi(X)^2$ and $\mathbb E_\psi(B^2)=\Delta_\psi(P)^2$. Applying the preceding inequality and taking nonnegative square roots proves the <Heisenberg uncertainty relation>:
$$
\Delta_\psi(X)^2\Delta_\psi(P)^2\geq\frac{\hbar^2}4,\qquad \boxed{\Delta_\psi(X)\Delta_\psi(P)\geq\frac\hbar2.}
$$
\b[Subtracting the means is essential]: the uncentred second moments alone do not equal the <variances>. The <equality case of the Heisenberg uncertainty relation> requires the centred vectors to be imaginary multiples, as in the preceding equality conditions.
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