Uncertainty product of oscillator number states
= Uncertainty product of oscillator number states
An <energy eigenstate> has $\langle x\rangle=\langle p\rangle=0$, $\langle x^2\rangle=\hbar(n+1/2)/(m\omega)$ and $\langle p^2\rangle=m\hbar\omega(n+1/2)$. Thus $\Delta x\Delta p=\hbar(n+1/2)$; only the ground state in this sequence minimizes the <Heisenberg uncertainty principle>.