Why is it that we cannot both know the momentum* and* position of a sub-atomic particle?
Okay, I think I get the general idea. The more you focus on just one aspect of a sub-atomic particle the less you are able to define another. You have to resolve one or the other.
Thank you. 
On a mathematical level, it’s a property of waves. Once you have a wave equation governing the system, you are going to get properties like this. Energy and Time work the same way as position and momentum.
Derleth, that was a beautiful explanation. Of course, I understood well all the mathematical principles involved, but had never related them to the physics. But the drum was especially convincing.
Some day I’d like to understand what Hilbert space has to do with QM.
Well-written post!
I first encountered this idea in a discussion of signal (or image) analysis. A wavelet transform representation was described as a compromise between spatial domain (pixels) and frequency domain (Fourier), giving information about each. The computer scientist discussing this was also a professor of physics, so naturally mentioned the connection to Heisenberg’s Principle!
Thank you, and it’s good to see another example of this concept in use. I always love when seemingly unrelated fields are related in some deep, interesting way, as opposed through sheer trivia.
Thank you. I always try to have something specific to use as an example, if possible.
As for Hilbert spaces… I’m working on it, now that you’ve made that post. I can explain what they are, but as to why Hilbert spaces in specific are used as the phase spaces for quantum formalisms, I fear I’m still missing a big picture. They do have some nice properties, though.
Hang on…so now we’re saying sub-atomic particles are waves?? 
You might find this lad interesting.
Not many people can claim to have got a Nobel Prize for their PhD work, but he did.
Interesting, although, I didn’t understand a lot of it. So, when we try to measure a an ‘object’ at the atomic level are we measuring a particle or wave?
Both. But you don’t get to measure both perfectly. You only get so much information, and you can distribute it between knowledge of the wave and the particle. In principle you could make it 100%/0% one or the other, but in reality the split is somewhere between. It is possible to decide which side to put your measured information, and design the measurement process accordingly, something that can be very useful when pushing the limits.
Be clear that this isn’t just about measurement. It is an intrinsic part of how the universe works.
What it means however is a whole new can of worms. But the mathematics describing how this all hangs together predicts the experimental results to uncanny accuracy.
If we can measure something as either a particle or wave does that mean we can also measure things like baseballs, elephants, people, planets, etc. the same way?
Yes. But the wavelength is inversely related to the momentum and baseballs etc. have a lot more momentum than an electron, so the wavelength is insignificant.
But at smaller scale you can run the double slit experiment both with electrons and “large” molecules.
To get the wavelength of anything you take the momentum and divide by Planck’s constant, so a baseball fastball, for instance has a wavelength that’s about a billionth, billionth, billionth, billionth, billionth meter, give or take a few zeros since I didn’t do the calculations.
You must have missed Chinatown. Watch this excellent movie It has several memorable scenes but the climax is Scene 47:
Faye Dunaway’s character:
“She’s a particle.”
[Jake slaps her face]
“She’s a wave.”
[Jake slaps her other cheek]
“She’s a particle!”
[slapped]
“She’s a wave!!”
[slapped, breaks into uncontrollable sobs]
“She’s a particle and a wave!”
When you say a baseball has a lot more momentum than an electron I’m not quite sure what you mean. If a baseball is at rest doesn’t it have zero momentum?
Okay, but what about a baseball at rest?
With the indeterminacy principle there’s no absolute “at rest”. You can determine that the momentum is below some threshold, but the lower that threshold is, the less certain the position of the baseball becomes. I.e. the wavelength increases.
If you do the actual math you’ll find that for a baseball you can say that the baseball is moving slower than some ridiculously low speed, like a picometer per second, and the wavelength is still extremely small.
For an electron, not so much.
I’m still a bit unclear.
So the faster an electron or a baseball is moving the greater the wavelength? Or isn’t it as simple as that?
The opposite. The faster it goes (or the heavier it is), the shorter the wavelength.
The problem with an object at complete and total rest is that its wavelength is infinite*. Which is not possible. Hence everything is always jiggling around in a certain sense. The bigger the object, the less the “jiggle”. As noted, a baseball is so immense in terms of quantum effects it’s impossible to realistically measure such motion.
- Which means its position could be nailed down to “Somewhere in the Universe.”