# Standard Deviation of Position and Momentum

Classical Mechanics Level 5

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 $$a$$.

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