# Uncertainty Principle Upper-bound?

In quantum mechanics, is there an upper bound for the uncertainty principle? I know that quantum harmonic oscillator (QHO) has the uncertainty relation $\sigma_x\sigma_p = \hbar(n+1/2)$, but I think the QHO becomes localized at two peaks spread out over a large distance?

-

When you have two operators $\hat{A}, \hat{B}$ satisfying equation $[\hat{A}, \hat{B}] = \imath \hat{C}$, you can prove with Schwarz inequality that $\sigma_{\hat{A}, \psi} \sigma_{\hat{B}, \psi} \geq \frac{1}{2} | \hat{C} |$. Unless there would be stronger inequality that can be used in calculations, it gives us the lower bound of uncertainty principle.