To the best of our understand, particles are fundamentally wavelike in nature. For example, if you shoot electrons through closely spaces slits, they will form interference patterns just like light does (There are lots of other experiments you can do to demonstrate this as well, but this is perhaps the most straightforward one). So, if the wavelike nature of particles is fundamental (which we believe to be true), then the Uncertainty Principle is also fundamentally true. In fact, the Uncertainty Principle can be derived, without any reference to “measurement” at all.

There are actually many different uncertainty relations- the Heisenberg Uncertainty Principle being the most famous one- that there is an uncertainty between the position and momentum of a particle. But really, any two observables (observable being a quantum mechanics word for “something you can measure”) which do not commute will have an uncertainty relation. What does this mean? So, when something commutes, it means order of operations doesn’t matter. For instance, A + B = B + A. Addition commutes. Multiplication sometimes commutes. For instance, if x and y are just numbers, then xy = yx. But, if x and y are matrices, then x*y ~= y*x. In Quantum Mechanics, operators (or functions which operate on the wavefunction) sometimes commute and sometimes don’t. Ones that don’t (like position and momentum), will always have an associated uncertainty principle.

Position and Momentum are part of the canonical commutation relation. This means if the position operator (P) operates on the wavefunction (W) first, and then the momentum operator (M), you get a different answer than if the momentum operator operates first, and then the position. Or in math: [XP – PX]*W = i*h_bar, where i is the imaginary number, and h_bar is Plank’s constant divided by 2*pi. Another common pairing that shares this relationship is Energy and Time, thus they also have an uncertainty principle.

While perhaps this got pretty far into the weeds, the Wikipedia article summarizes it nicely:

> Historically, the uncertainty principle has been confused with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the system, that is, without changing something in a system. Heisenberg utilized such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty. It has since become clearer, however, that the uncertainty principle is inherent in the properties of all wave-like systems, and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology. Indeed the uncertainty principle has its roots in how we apply calculus to write the basic equations of mechanics. It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.

An analogy I've found useful: where is a 1 Hz sinusoidal wave? Well, it's everywhere. Having a precise frequency goes hand in hand with extending from -∞ to ∞. It has no single location.

What's the frequency of a point? Well, it doesn't have one; there's no physical extent for us to examine its periodicity.

In between these two extremes, you could have a localized wave whose position can't be well defined because it's not pointlike. Its frequency also can't be well defined because it's not perfectly periodic. You could estimate these two values, but they'll always contain ambiguity; in fact, as one becomes more certain, the other becomes less certain. This has nothing to do with a measurement limitation. It's a fundamental constraint.

This is unfortunately caused by the pop-science presentations of the uncertainty principle. To answer this fully, two different pieces have to be teased apart, and precision in language becomes important. There is a general confusion of two different phenomena. The uncertainty principle and the observer effect.

Observer Effect A phenomenon where measuring a system requires interacting with that system and changes the observables of that system.

The uncertainty principle Given an experiment where we can set the initial position of a particle to be at (0x, 0y, 0z, 0t) with a momentum p and we later measure the position and momentum of that particle a time t later, if we repeat that experiment for many particles, we will get a range of different final positions and momenta. The uncertainty principle holds that the product of the standard deviations of the distributions of final position and momenta, given a definite starting position and momentum, cannot be less than some limit. If one knows a particle's position and momentum now, one cannot predict both what its future position and momentum will be.

These two ideas (the uncertainty principle and the observer effect) are completely unrelated. Just because when a system is measured, it has to be interacted with does not mean that interaction is unpredictable. We can, in principle, fully account for all interactions and know how the measurement will affect the system and back out information about the system pre-interaction.

The uncertainty principle is orthogonal to measurement, and is related to what is possibly knowable about a system given full, perfect information. We can know perfectly where a particle is now, and what its momentum is now, and that will not allow us to accurately predict what its position and momentum is in the future.

The uncertainty principle is epistomological, it is a principle about what is fundamentally knowable. We can know everything there is to know about a system, we can fully specify its wavefunction and know how it will evolve forever and still not be able to predict the values of all observables about the system in the future. Knowledge of the future position of a particle is fundamentally incompatible with knowledge of its future momentum. Joint knowledge of future position and momentum are guaranteed to always be fuzzy with some spread in either or both observables.

There are often things reported about measurements "beyond the Heisenberg limit" which are breathlessly reported on, saying that they "broke the uncertainty limits" or whatever. Those are related to incorrect statements about the uncertainty principle and that it applies to any one measurement at all. To be clear, the uncertainty principle does not restrict what one can measure about a particle. You can perfectly measure a particle's position and momentum at any given time. The thing that the uncertainty principle forbids is using that information to know, with certainty, the future position and momentum of the particle.

> But as far as I'm concerned that is not a natural law but a restriction caused by our own inability to observe those data points without influencing the system ourselves.

To be very clear, it is a natural law that is not premised on experimental failures to observe things nor is it caused by our inability to observe without influencing. In mathematical models where all information about a system is known precisely, and all measurements are perfect, the results are the same. Future position and momentum cannot be known with joint precision below a certain limit.

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There are some damn fine answers here. Lol.

Short answer is it's a natural law that has been proven.

Additionally our experiments have limits because they are imperfect and restricted by power inputs and financial resources. Improvement will only help us get closer to the max allowed by the uncertainty principle.