Why is it impossible to measure position and momentum at the same time with arbitrary precision?

Question

I'm aware of the uncertainty principle that doesn't allow $\Delta x$ and $\Delta p$ to be both arbitrarily close to zero. I understand this by looking at the wave function and seeing that if one is sharply peeked its fourier transform will be wide.

But how does this stop one from measuring both position and momentum at the same time? I've googled this question, but all I found were explantions using the 'Observer effect'. I'm not sure, but I think this effect is very different from the intrinsic uncertainty principle.

So what stops us from measuring both position and momentum with arbitrairy precision? Does a quantum system always have to change when observerd? Or does it have to do with the uncertainty principle?

Thank you in advance.

This post imported from StackExchange Physics at 2025-03-08 12:38 (UTC), posted by SE-user catmousedog

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This post imported from StackExchange Physics at 2025-03-08 12:38 (UTC), posted by SE-user rob

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Since it was deleted, here's 3blue1brown's explanation of the general uncertainty principle. The tl;dw is that the "uncertainty principle" is actually a property of all waves.

This post imported from StackExchange Physics at 2025-03-08 12:38 (UTC), posted by SE-user BlueRaja - Danny Pflughoeft