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I know that the uncertainty principle is: $$\Delta p\Delta q \ge \frac{\hbar}{2}.$$

But do the units on the left-hand side of the equation always have to equal $\text{Js}$, i.e. $\text{energy} \times \text{time}$ (the same as Planck's constant) or is it simply the numerical value which matters in the inequality.

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    $\begingroup$ The uncertainty principle is no different from any other physical equation. Unless a specific system of units is being specified, the dimensions of each side should match. $\endgroup$ – Javier Dec 14 '14 at 21:16
  • $\begingroup$ Related: physics.stackexchange.com/q/28957/2451 and links therein. $\endgroup$ – Qmechanic Dec 15 '14 at 0:53
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The uncertainty principle may be stated more generally for two observables $A$ and $B$ as $$ \Delta A \Delta B \geq \dfrac{1}{2}\left|\langle\left[\hat{A},\hat{B}\right]\rangle\right|, $$

where $\langle \hat{C}\rangle$ is the expected value of the observable $C$ and $[\cdot\,,\cdot]$ is the commutator (see here for details). From this equation, we can see that the units of both sides are automatically the same (i.e., both sides have the units of $A$ multiplied by the units of $B$).

In the case of momentum $P$ and position $Q$ (using your notation), one can show that $\left[\hat{P},\hat{Q}\right]=-i\hbar$, which, substituted into the previous equation, gives the uncertainty principle given in the OP.

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I'm essentially interpretting your question as

"are there any canonical commutation relationships (CCRs) where the Lie bracket is a different scaling constant of the identity matrix:

$$[\hat{x},\,\hat{p}] = i\,\alpha\,\mathrm{id}\tag{1}$$

or, equivalently, are there any pairs of canonically commuting observables where the scaling constant $\hbar$?"

Naturally, the units of $\alpha$ must match the product of the units of the canonically commuting variables (CCVs). So the question becomes: "what is the dimension of $\hat{p}\,\hat{q}$, where $\hat{p}$ and $\hat{q}$ are CCVs.

We can think of CCVs as coming from their classical counterparts in Hamiltonian mechanics, where conjugate variables are defined by:

$$\begin{align} & \frac{d\boldsymbol{p}}{dt} = -\frac{\partial \mathcal{H}}{\partial \boldsymbol{q}}\\ & \frac{d\boldsymbol{q}}{dt} = +\frac{\partial \mathcal{H}}{\partial \boldsymbol{p}}\end{align}\tag{2}$$

and the Hamiltonian evolution equation in QM is gotten by applying "Dirac's Quantisation Rule" of replacing Poisson by Lie brackets:

$$\mathrm{d}_t \hat{p} = -\{\{\hat{p}, \hat{H}\}\} + \partial_t \hat{p}\quad\text{ becomes }\quad \mathrm{d}_t \hat{p} = -\frac{i}{\hbar} [\hat{H}, \hat{p}] + \partial_t \hat{p}\tag{3}$$

to get the time evolution equation for observable $p$ from its corresponding evolution as defined by the classical Hamilton's equation. I say more about all this in my answer here (see especially the historical papers I link to in that answer).

(2) shows that candidate conjugate variables must have the dimensions:

$$[\hat{p}][\hat{q}][t^{-1}] = [\mathcal{H}]$$

and thus that the dimension of $[\hat{p}\,\hat{q}]$ must be the dimensions of the Hamiltonian times time. (3) shows that the Hamiltonian's dimensions must fulfill:

$$[\hat{H}]=[t^{-1}][\hbar]$$

and so that the dimensions of the right hand side in the CCRs must be the same always as that of Dirac's constant $\hbar$ if we accept Dirac's quantisation procedure.

Not all commutation relationships are like this in QM. The CCR above arises between observables defined on noncompact spaces with nondiscrete spectrums (the CCR essentially asserts that a quantum state in one conjugate variable's eigenco-ordinates is the Fourier transform of the quantum state expressed in the other conjugate variable's eigenco-ordinates). Angular momentum and spin observables are compact operators and as such fulfill relationships between the three Cartesian components like:

$$[{L_x}, {L_y}] = i \hbar \epsilon_{xyz} {L_z}$$

Some other CCRs that may not be wonted to you are corresponding components of the electromagnetic field vectors in the Quantized description of the Electromagnetic Field:

$$[\hat{A}_j,\,\hat{J}_k] = i\,\hbar\,\delta_{j\,k}\,\mathrm{id}$$

where $\hat{A}$ is the vectormagnetic potential and $\hat{J}$ the current density.

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  • $\begingroup$ WetSavannaAnimal aka Rod Vance: Comparing the first and last equation of your answer I wonder whether your notation is consistent, i.e. in terms of the index which only appears in the latter eq. Otherwise +1. (Also, I've done a bit of copy-editing; still struggling hard to plug my assorted reputation leaks ... &) $\endgroup$ – user12262 Jul 3 '15 at 5:18
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I now know (as pointed out by 'Javier Badia') that as this inequality is like any other normal inequality, the units on both sides of the inequality must be equal.

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