Uncertainty Principle in 3 dimensions

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?

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.

• 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? Mar 29 at 18:20
• @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. Mar 29 at 18:47

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) $$\Delta x^i \Delta p^i \geq \frac{1}{2}\hbar$$.

• What kind of inequality is that? Aug 2 at 4:59

The uncertainty relation holds for each direction in space-time, so $$\Delta x_i \Delta p_i \geq \hbar/2$$ for $$i=0,1,2,3$$ ($$t,x,y,z$$).