After a good chunk of time today working with natural units on another project, and later walking and pondering on the uncertainty relations from quantum mechanics: is there any fundamental reason that the units of the uncertainty relation should be the same units of action? I understand that the uncertainty relations all end up being $\propto \hbar$, (e.g. $\Delta E \Delta t \propto \hbar$) but is there any fundamental reason why this must be? Are there any uncertainty relations that do not have units of action?
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1$\begingroup$ Essentially a duplicate of physics.stackexchange.com/q/28957/2451 $\endgroup$– Qmechanic ♦Commented Jun 8, 2017 at 11:22
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Actually since $\Delta A\Delta B\ge \frac{1}{2}\vert \langle [\hat A,\hat B]\rangle\vert$, it is entirely possible for the product of uncertainties to have dimension other than action. For instance, if $\hat A=\hat L_z$ and $\hat B=\hat x$ then $$ \Delta L_z\Delta x\ge \frac{\hbar}{2}\vert\langle \hat y\rangle \vert $$ and so the uncertainty would have units of [action]$\times$[position]. Likewise, $\Delta L_x\Delta L_y\ge \frac{\hbar}{2}\vert\langle \hat L_z\rangle\vert$ would have units of $\hbar^2$ since $\langle \hat L_z\rangle$ will have units of $\hbar$.
answered Jun 8, 2017 at 11:17