 Solution

The expectation value of an observable is the mean outcome of repeated measurements on identically prepared systems in the normalized state. For a self-adjoint $Q$ it is real and equals $⟨\psi ,Q\psi ⟩$. Assume the state belongs to the domains needed for the second moments and the commutator below, so all these quantities exist.

For real $\lambda$, positivity of the squared norm givesThe commutator expectation is purely imaginary, so this is a real quadratic in $\lambda$. If $⟨P^2⟩>0$, its minimum is nonnegative exactly when its discriminant is nonpositive. If $⟨P^2⟩=0$, nonnegativity for all real $\lambda$ forces the linear coefficient to vanish, yielding the same conclusion. ThusReplace the observables by their centred versions $Q-⟨Q⟩I$, $P-⟨P⟩I$. Their commutator is unchanged, and their second moments are the variances $(\Delta Q{)}^2,(\Delta P{)}^2$. Taking square roots gives the Robertson uncertainty principleFor the quantum harmonic oscillator, $p=-i\hslash {\partial }_x$ gives $[x,p]\psi =i\hslash \psi$. The uncentred inequality above consequently gives $⟨x^2⟩⟨p^2⟩\ge {\hslash }^2/4$. The arithmetic-geometric mean inequality now yieldsThis is a bound for every normalized state of finite oscillator energy, not just an eigenstate. It is attained by the centred Gaussian ground state, for which $⟨p^2⟩=m^2{\omega }^2⟨x^2⟩$ and the uncertainty product equals $\hslash /2$.