 Solution

For a normalized state and a self-adjoint operator $Q$ with finite second moment, the quantum uncertainty is defined by $\Delta Q=\Vert (Q-⟨Q⟩)\psi \Vert$. Since $⟨Q⟩$ is real,In an energy eigenstate of the quantum harmonic oscillator,Discarding the nonnegative squared means proves the stated lower bound. The Heisenberg uncertainty principle is $\Delta x\Delta p\ge \hslash /2$, following from $[x,p]=i\hslash$. Applying the arithmetic-geometric mean inequality to the two variance contributions givesHere $\omega >0$ is the oscillator frequency. The bound is the ground-state energy; equality requires zero means and the appropriate minimum-uncertainty width.