Standard Deviation of Position and Momentum

Let \[\psi(x) = {\pi}^{-1/4} {e}^{-{x}^{2}/2}.\]

First calculate \({\sigma}_{x} = \sqrt{\left<{x}^{2}\right> - {\left<x\right>}^{2}}\).

Then calculate \({\sigma}_{p} = \sqrt{\left<{p}^{2}\right> - {\left<p\right>}^{2}}\).

If \({\sigma}_{x}{\sigma}_{p} = a\hbar\), evaluate \(a\).

Image Credit: Wikipedia Fermilab, U.S. Department of Energy
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