= Harmonic-oscillator number-state second moments
{title2=$\langle x^2\rangle_n=\hbar(n+1/2)/(m\omega),\quad\langle p^2\rangle_n=m\hbar\omega(n+1/2)$}
For a <number operator> <eigenstate> $|n\rangle$, the <annihilation operator> and <creation operator> satisfy $\langle a\rangle=\langle a^\dagger\rangle=\langle a^2\rangle=\langle(a^\dagger)^2\rangle=0$, $\langle a^\dagger a\rangle=n$ and $\langle aa^\dagger\rangle=n+1$. Inserting $x=\sqrt{\hbar/(2m\omega)}(a+a^\dagger)$ and $p=i\sqrt{m\hbar\omega/2}(a^\dagger-a)$ gives the displayed second moments and zero means.
Thus the <kinetic energy> and <potential energy> each have mean $\hbar\omega(n+1/2)/2$, and the uncertainty product is $\Delta x\Delta p=\hbar(n+1/2)$. Among number states only the ground state saturates the <Heisenberg uncertainty relation>. A <quantum superposition> can have nonzero position and momentum means despite each constituent number state's zero means.
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