How do first-principles help understand uncertainty in quantum physics?

Let's put it this way: first principles themselves are not a tool to "eliminate" uncertainty. Quite the opposite, they "predict" the inevitable existence of uncertainty from the most fundamental level.

You can imagine this as playing a game.

First principles are like the "underlying rulebook" of this game. In the game of quantum physics, the core rulebook consists of things like the Schrödinger equation. When we conduct research, we must strictly adhere to this manual, not invent rules, nor rely on intuition to say, "I think it should be this way."

Uncertainty, then, is a "peculiar phenomenon" you discover while playing this game.

Now let's connect the two:

When physicists open the "rulebook" (i.e., apply first principles) to describe the simplest particle, such as an electron, the manual doesn't state "this electron is at point A with velocity B"; instead, it gives you something called a "wave function."

This "wave function" is closer to the electron's true nature; it's not a point, but a "probability cloud." The area covered by this cloud represents the possible locations of the electron; and the way this cloud fluctuates relates to its momentum (which you can roughly understand as velocity).

The key point is this: The very nature of the "cloud" inherently carries uncertainty.

1. If you want the position of this cloud to be very certain, you have to "squeeze" it very small, almost shrinking it into a point. However, according to the mathematical calculations in the "rulebook," a "probability cloud" squeezed so small will inevitably have extremely chaotic and complex fluctuations, encompassing a wide range of momenta. Therefore, its momentum becomes highly uncertain.

2. Conversely, if you want the momentum of this cloud to be very certain, meaning its fluctuation pattern is very pure and singular. Then, according to mathematical calculations, such a "cloud" will inevitably spread out very widely in space, extending over a long distance. Therefore, its position becomes highly uncertain.

You see, when you strictly adhere to the "first principles" rulebook to describe a particle, you find that the two properties, "position" and "momentum," are like the two ends of a seesaw: if you push one end down firmly (making it very certain), the other end will inevitably rise high (making it very uncertain).

Therefore, first principles don't help us "understand" uncertainty; rather, they tell us from the most fundamental level: uncertainty isn't due to poor measurement techniques or a lack of understanding on our part; it is a fundamental property of the quantum world, embedded in its very rules. As long as you accept that "rulebook" as correct, then this phenomenon of "uncertainty" is an inevitable and inescapable conclusion.