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I'm trying to understand how to write Heisenberg uncertainty principle in 3 dimensions. What I mean by that is to prove something of the form $f(\Delta p_x,\Delta p_y,\Delta p_z,\Delta x,\Delta y,\Delta z) \geq A$.

This is what I got: The unknown volume that a single particle can be in is $\Delta V=\Delta x \Delta y \Delta z$. The uncertainty in the size of the momentum is $\Delta p = \sqrt{\Delta p_x^2 + \Delta p_y^2 + \Delta p_z^2}$

Now this is where I get stuck. In my textbook, for the 1d case, they used De-Broglie equation for connecting the uncertainty of the particle wavelength and its momentum along the $x$-axis. But does De-Broglie equation is correct per axis or for the size of the vectors?

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3 Answers 3

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If you choose any direction in space, the 1D uncertainty principle applies in that direction. So, if you know the component of a particle's momentum in that direction well, you cannot know its projected position in that direction well. The directions of good and poor localization need not be aligned with your coordinate axes.

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  • $\begingroup$ If I take choose two non parallel directions than wouldn't that mean the I can know the exact location on those two axis of the particle? $\endgroup$
    – Yotam Ohad
    Mar 29 at 18:20
  • $\begingroup$ @YotamOhad No. Suppose you have a bound on Δ𝑥 based on Δ𝑝𝑥. That tells you nothing about Δy or Δz. Knowledge along one axis doesn't mean that the particle is localized on that axis. $\endgroup$
    – John Doty
    Mar 29 at 18:47
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There is no need for a 3D uncertainty principle.

The operators commute between dimensions:

$$[\hat{x}^i, \hat{p}^j] = i \hbar \delta^{ij}$$

Momentum in one dimension and position in another can be measured with arbitrary precision simultaneously (at least there is no restriction from QM).

The closest thing I can think of for what you're asking is:

$$ \sum_{i=1}^{3} \Delta x^i \Delta p^i \geq \frac{3}{2}\hbar $$

which is a weaker inequality than (and not as useful as) $\Delta x^i \Delta p^i \geq \frac{1}{2}\hbar $.

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  • $\begingroup$ What kind of inequality is that? $\endgroup$ Aug 2 at 4:59
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The uncertainty relation holds for each direction in space-time, so $\Delta x_i \Delta p_i \geq \hbar/2$ for $i=0,1,2,3$ ($t,x,y,z$).

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