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I disagree with the first bullet point. If there was no such a thing, how could we even prove the second point? What I mean is that one has to measure both the position and momentum of a particle many times if one aspires to fill a histogram that may resamble a PDF.
If you confine a particle to go through a tiny slit, the more spread you will have in the momentum distribution. Take a photon, as it goes through the slit you will precisely where it is but you won't know the direction it is headed (i.e. it's instantaneous momentum vector)
Make the slit smaller and smaller, the light the resulting interference pattern will grow wider and wider. If you can get both at the same time then you my friend will get a nobel prize
Again, look at how you would measure the interference pattern. I think that one would throw one photon at a time. And for each photon, one would need to know both the slit size and the place where it lands. Effectively measuring both x and p.
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