Heisenberg uncertainty principle during measurement

Question

I am currently studying quantum mechanics and again came across the uncertainty principle, and I seem to lack any intuition about what it means. I full understand how it can be derived using commutators or Fourier transforms. However I am unsure how to interpret the results. As I understand, this result is a mathematical consequence of one of the axioms. I now have 2 questions:

1. How was this principle experimentally verified or is it now also part of the axioms - meaning there is no way to verify but it seems to describe reality as we observe it.
2. How does the principle work when taking a measurement ? Suppose I put an electron into an electric field where the total force on the electron is zero if it is the exact middle of the field. Then I know if the electron doesn't move it has to be in the very middle. So the location and velocity are now linked, but this should not be allowed right?

I am very new to this forum so let me know if the question needs adjustment :)

Answer 1

Well, we have an electron trapped in the exact middle of an electric field: the hydrogen atom.

It doesn't plant itself in the middle of the nucleus (nor does it fall to the bottom of the infinite potential well in the point charge approximation--or positronium if you want to keep it experimental). Rather, the Bohr radius is consistent with the uncertainty principle and momentum of the ground state (https://quantummechanics.ucsd.edu/ph130a/130_notes/node98.html)

Regarding experimental verification of a saturated position/momentum uncertainty principle with an single electron in the lab: I don't know what the measurement limit is.

Other experimental verification are:

1. observations of quantized $z$-axis angular momentum and the uncertainty of projections along the $x$ and $y$ axes.

2. Diffraction of electron beams/photons through a narrow slit, where the uncertainty in the transverse momentum is directly related to finite width of the slit via HUP.

3. Squeezed light, but I am not an expert in quantum optics.

A final comment, that is just an opinion, is that I find the difficulty in intuitively understanding the position/momentum version of HUP is our classical intuition as momentum being $p=mv$, which is well baked by the time we get to quantum mechanics.

Instead, start with Noether's theorem and translation operators to find:

$$ \hat p = -i\hbar \frac{d}{dx} $$

and consider how that operates on a wave-packet... remembering that most of the derivate comes from the phase, not the amplitude envelope. I've seen that in many pop-sci explanations of the HUP, presenters only show the real part of the wave function, and at that point: all is lost.