system
Given the following post, respond as if you're commenting on Reddit:
human
Why does the Uncertainty Principle necessitate randomness in the universe and not just a limit to our ability to measure?

I'm reading A Brief History of Time, and one of the topics that's really not registering with my brain is this Uncertainty Principle. As I understand it, when we want to measure the state of a subatomic particle (let's just use an electron because that's what my boy, Stevie H. used), we want to figure out it's location and velocity.

To do that, we shine light at it (and need to direct at least one quanta of light). We know that the higher the frequency of the light we shine, the clearer the position of the particle will be for us (because we can only be as precise as one wavelength), but then the higher the frequency of the light, the more energy it stores and the more disturbance is imparted on the particle.

So if I have that (mostly) straight, what I don't understand is the conclusion. Can we not imagine that Laplas' Demon may still have a way of observing the particle without disturbing it? Isn't this more a problem of our own limitations as observers than it is an inherent truth in the universe?
gpt
Forget about whether things are "really" waves, particles, or some sort of duality. That's not important. What *is* important is that we can *represent* the state of a system in various mathematical ways, some of which are more "wave like", some of which are "particle like", and some of which incorporate a bit of both. That is, we can write down mathematical expressions that relate varies properties of the system, and in some cases those expressions look like the ones we would use for waves while in other cases they look like the expressions we would use for, say, a baseball.

Now, we look at how some system behaves and we come up with a way to represent that behavior mathematically. Then we can ask "what does this mathematical representation *imply*?" The representation was created to describe some specific phenomenon, but perhaps we can vary it or generalize it a bit to describe other phenomena. Then we can see whether the expression accurately predicts the results of various experiments.

So that's the general idea, and we ran through that process with the behavior of very small bits of stuff. Like single atoms of silver, for instance. What came out of it was the mathematical formalism we use to describe quantum mechanics. We then tested that formalism in a large number of different scenarios, and it accurately matched most of them (the ones it didn't turned out to require relativistic corrections, because the bits were moving very fast, for example). But, it had some other interesting consequences. One could show *from the formalism* that there was an *intrinsic* relationship between the uncertainty in position and the uncertainty in momentum for a system. In other words, you can represent the state of a system in terms of its momentum or its position, but a single state cannot be represented by both a single-valued position and a single-valued momentum. In fact, the more "narrow" the region to within which its position can be constrained, the more "broad" the values of momentum that it could possibly have, which means that you can't represent it by a single-value position *at all*—there must always be some finite-sized region over which it is localized.

So the statement is: "If this mathematical formalism accurately describes the behavior of quantum systems, then there is an intrinsic uncertainty in the momentum and position of the system; the more precisely one is determined, the less precisely the other is determined *even in principle*." Since the mathematical formalism *does* accurately describe quantum systems, we conclude that there is such an intrinsic uncertainty.