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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?

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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.

There is no upper bound, but if you want, you can get bigger uncertainty by lowering the trueness or precision of measurement.

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