 Solution

For a normalized wavefunction, $⟨A{⟩}_{\psi }$ is the statistical mean of measurements of the quantum observable $A$ over identically prepared systems, not necessarily the outcome of one measurement. Assume the state is in the required operator domains and the second moments are finite. Hermiticity givesThe commutator expectation is purely imaginary, so $i⟨[A,B]⟩$ is real. A real quadratic nonnegative for every $\lambda$ has nonpositive discriminant; if its quadratic coefficient vanishes, its linear coefficient must also vanish. ConsequentlyApply the same inequality to $A-⟨A⟩I$ and $B-⟨B⟩I$. Their commutator is unchanged and their second moments are the variances, yielding the quantum uncertainty relationFor the harmonic oscillator, $p=-i\hslash {\partial }_x$ gives $[x,p]\psi =i\hslash \psi$ by the product rule. The uncentered inequality above therefore implies $⟨x^2⟩⟨p^2⟩\ge {\hslash }^2/4$. The arithmetic-geometric mean inequality now givesNo assumption that the position or momentum means vanish is needed.