OK, deleted a wall of text that I could not keep from getting ridiculously technical for this forum.


Accept my stipulation that all particles obey the Schroedinger Equation.
Accept that solutions to the Schroedinger Equation are functions, and that those functions are identical to wave-functions.

Everything that applies to waves applies to the solutions of the Schroedinger Equation. And those solutions describe every predictable property that every particle has.

Trust me on this.


OK. Let's just talk about waves.

You are standing near a pier, on a beach. You take a photo of the waves along the side of the pier. You know the pier is 100 feet long. In the photo, you see the waves rolling to the shore frozen in still frame.

Using cleverness, you decide to figure out the wavelength of the wave. Knowing the pier is 100 feet long, you count the number of peaks along the pier. You count 10 peaks. (I'm just throwing out arbitrary numbers. I live thousands of miles from an ocean.)

So you estimate the wave to have a wavelength of 100 ft / 10 = 10 feet.
If you're being rigorous about your observation, you may say the peaks are hard to measure precisely, and while there are 10 peaks near the pier, there may be 9.5 to 10.5 peaks, for really. Call it 10.0 +/- 0.5. The +/- 5 is the measurement uncertainty, which is another layer of uncertainty on top of the observer effect and the HUP.

Now. That's a pretty good measure of the wavelength. We can directly relate that wavelength to the frequency if we know the wave's speed. We can directly relate the frequency to the energy with a bit more data. So ultimately, by measuring the wavelength, we're measuring the Energy. All of this is kinetic energy, so we're really measuring the momentum of the wave.

So we have a fairly well defined momentum.

Now.

What is that wave's position? Tricky, right?

I mean.. we used information from all along that 100 foot pier to establish the momentum. So the location is, at best, somewhere along that 100 foot pier.


Now suppose you see a single pulse of a wave going down a string. What is the position of the pulse?
Well, that's pretty well defined. Not great, but much better than the many-peaked wave by the pier.

Well... what's the wavelength of the pulse?
Uhh... There's only one peak... how am I supposed to measure the wavelength, right?

Right.


And while there is uncertainty in our measurement, it is wholly separate and above the already-present uncertainty in these properties of the wave.