What is The Heisenberg Uncertainty Principle Making Statements About?

Question

Non-physicist here, trying to understand details about Heisenberg's uncertainty principle:

Watching the The more general uncertainty principle, beyond quantum about the uncertainty principle I came to the understanding that the uncertainty principle is mathematically highly linked to the general uncertainty principle.

My understanding is, that the Heisenberg uncertainty principle is the more complex application of that purely mathematical principle to quantum-particles. Since the current physical quantum model uses wave-functions to describe these, it is quite similar to the general mathematical uncertainty principal + some workarounds for quantum-specific stuff I don't understand.

Just to recap my understanding of the mathematical “problem” in my own words: if we have a wave-function the FFT-based frequency “analysis” is most precise when integrating an infinitely long sample while the “event time” is most precise when the sample is infinitely small/short.

Now what I am unsure of (assuming my understanding so far isn't way off): Does the Heisenberg uncertainty principle make a statement about our (mathematical) probability/wave-function-based model of quantum-particles or the “actual” [edit because of confusion: Does it try to model/describe reality] quantum-particles?

Let me rephrase: Is the proof linked with/based on the wave-function such that if humanity was to find a more accurate model of quantum particles in the future which wouldn't be based on wave-functions or probabilities, would the Heisenberg uncertainty principle still hold up for the new model or would it have been a purely mathematical limit of the old model imposed by the use of wave-function/probabilities.

And, assuming my previous question made any sense: Would, therefore, any philosophical conclusions based on the Heisenberg uncertainty principle about our universe/particles be generally valid or using the principle outside its intended scope/meaning.

On a side note (all representing my understanding): While the original derivation and a lot of other papers (to me) seem to base all assumptions made on the used wave-functions a lot of more generic articles seem to argue that it is not related to the model used but a universal truth. I might just not get it, but most of the arguments feel a little cyclic to me, starting with a pattern similar to this:

1. Uncertainty principle is given/proven.

2. We can't use orbit-based models since those would violate 1 ⇒ use waves.

3. Since we have uncertainty and no orbits, we have to use probability.

My problem is probably me not understanding all the steps or premises correctly but it feels that in a lot of articles/papers a potential existence of a superior model that allows determinism is ignored or falsified by arguing that it would violate the uncertainty principle.

Answer 1

Insofar as your question regarding whether or not it is a statement about "actual" particles, as @knzhou suggests, we don't really have access to "actual" reality in general in any kind of science (or perhaps we do - this really more depends on your philosophical viewpoint). You see, a scientific theory is perhaps best understood as what I'd call a "useful story" about things: it's a way of thinking that the world is, that has the property that it lets us accurately answer questions regarding what the consequences are of things that we may choose to do in the sense that if we actually do those things then, (to the extent of "usefulness" of the theory) what it says will happen, happens. Presumably, it must thus capture some of the underlying "logic" of the "real" real world, but it may not necessarily do so in the same way as it "really exists", whatever that means. Hence the aforementioned comment.

However, I also think there's another way to interpret your question, that is more interesting to answer: that is, within the "story"'s reality, i.e. that of the "story of quantum theory", what is the status of HUP? In particular, in Newtonian mechanics, which is another "story", that is "less useful" in the sense I just mentioned in that it doesn't give us accurate consequential answers in all cases, the point-like or solid objects that are imagined as moving about according to Newtonian rules have the status within the theory of being "actual" objects moving around in the "real world", and their parameters like the position and velocity, likewise, are parameters we affix to them to precisely specify their arrangement in space and their future motions.

And thus, we may ask, what is the status of the wave function and HUP in quantum theory "best" likewise understood as? Is it understandable as being like the objects in Newtonian mechanics as just mentioned - with then the attendant consequence you run into the idea they are "mysteriously" somehow affected by an "observer" in an almost supernatural way? Now, I'm sure you know, many different ideas have been proffered about this, but I'd like to proffer another, which I believe cogently integrates all the ideas of quantum theory.

Quantum theory, basically, is actually a theory whose focus, or vantage point, is fully that of the "observer", which only makes some partial statements regarding the "outside universe" (within the theory). In particular, the famous "wave function", or more generally the quantum state vector, $|\psi\rangle$, is a mathematical model, not of the "physical object" in the outside universe itself, but instead of the "observer"'s - which here we will better call the agent's - possessed information regarding the parameters of, or better, questions that can be asked about an external physical object.

This point that it is a model of information is important - this, I believe, is one of the things that gets many tripped up: one should not expect a real agent to be literally storing a wave function, so questions like the strangely inordinate complexity of such are thus taken care of right there. The quantum agent is a kind of "idealized" agent, and realistic agents may/may not share all properties of such, such as and including how they actually store information. It is ideal in the same way the numbers, point particles, perfect geometric rigid bodies, etc. of Newtonian mechanics are ideal. Note that there is also no prescription that an agent has to be a human being or have "consciousness" - all it needs to be able to do is to do three things: to store information, to acquire new information about the outside world regarding questions, and to update the information store with answers. When information is updated following a question, the vector $|\psi\rangle$ changes, and the agent's "experience" consists of a sequence of such.

The reason why we need this "strange" formalism is, in fact, precisely because of what the Heisenberg principle you are talking about best seems to be about: it is a representation of an informational limit in the outside Universe. This should be evident because it contains a physical constant, $\hbar$, but the rest of the mathematical formalism does not. In fact, the textbook uncertainty principle is not strong enough - the formally stronger version is a truly informational statement expressed in terms of Shannon's informational entropy, and looks like this:

$$H_x + H_p \ge \lg(e \pi \hbar)$$

where we are measuring in entropic bits ($\lg$ is the binary, or base-2, logarithm). Informational entropy is a measure of the privation of information - it is how much information lacking regarding the answer to some question. Here, the relevant questions are "Where is the particle located?", which is represented by the position $x$ (or better, the position operator $\hat{x}$), and "How much momentum does the particle possess?", i.e $p$ (or $\hat{p}$, the momentum operator, sometimes also called the impulsion operator).

Now entropies are calculated from probabilities - and hence we find that we need a probabilistic framework, and this brings in the notion of an agent, probabilities as a way to mathematically represent possessed information, and the updating of the probabilistic information with new information (c.f. Bayesian theory). These probabilities get wrapped into the quantum state vector.

The principle, then, can be understood as saying that there is a lower bound on the privation of the total information an agent can have regarding both parameters: moreover, though, because it contains the physical constant, it is most reasonably understood as indicating a physical property of the outside Universe, and this property is a sort of "resolution limit", an upper limit on information content (lower bound on information privation becomes upper limit on information existence). In a sense, just like a computer game stores the position and speed of particles to a finite resolution, so too does the Universe in some sense, though the complexity of the probability distributions encountered in quantum mechanics means it can't be anywhere close to as simple as one in how it does so, so do not take that idea too literally. In this regard, $\hbar$ is analogous to $c$ in relativistic theory, there having a more well-understood interpretation as a limitation on the speed of information transport from one place to another. Hence, we have both a finite maximum speed of transportation of information through space, and a finite maximum quantity of information in any physical system - surely, this doesn't seem so unreasonable, now does it?

And the various other ideas come up with - "many worlds", "Bohmian mechanics" etc. are in this view better seen as different ways the "real" universe "could" implement, often with very heavy literal-taking of the quantum formalism's objects, particles in a way that their parameters would be informationally limited. But by this view, such are really a bit fruitless: it's essentially asking how the "game", so to speak, is implemented, and there are many equivalent ways to do that. Thus going back, again, to what I said about science more generally - it's our ideas of how to describe these things, and it should be taken as quite the irony that our best theory kind of reflects this very fact right back at us.

Answer 2

The uncertainty principle is a consequence of the mathematical structures underpinning quantum theory. Quantum theory is only an approximate model of reality, so we might possibly develop a more accurate model to which the uncertainty principle might not apply. However, the mathematics seem to suggest that there is a fundamental misalignment between the measurable properties of particles, so that an allowed value of one property, energy say, does not correspond to any specific one of the allowed values of another property, say position. If you measure the energy and get a specific value, when you then measure position you find a value on a probabilistic basis. But that value does not align with any particular energy value, so when you re-measure the energy you may get a different value from the original one. The particle simply cannot have a specific position and a specific energy at the same time. There is no exact analogue from everyday life, so it is difficult to explain the effect precisely without resorting to mathematics; however, the following analogy might give you an idea.

Suppose there is a menu with ten dishes and ten drinks. Some are more popular than others, but you can never be sure which combinations will be ordered. When someone orders their food they have to pick one of the allowed items; having done that they can pick any drink to go with it. However, someone else picking that same drink might not pick the same dish to go with it. There is simply no fixed correspondence between drinks and dishes. Some combinations might be more likely than others, but you can never predict exactly what combination a customer will order.

In the above example, the dishes might be the allowed values of energy and the drinks might be the allowed values of position.