Does the uncertainty principle go against chaos theory?

Question

My understanding of the uncertainty principle and quantum physics is that any given object may, without notice or explanation, spontaneously perform an action it previously was unable to do with a probability that a certain outcome will occur. However, chaos theory begs to differ in the fact that, given every bit of information about a certain scenario, a future outcome is fully predictable (through rigorous mathematics) with the slight of hand that the slightest change in initial conditions can have dramatically extreme affects on the future outcome.

So according to quantum physics, things can happen randomly.

According to chaos theory, everything is predictable, just extremely difficult.

Do the conflict or is my understanding just wrong?

Answer 1

Your question or confusion is mostly based on several misconceptions of the premises:

• Chaos theory is not a theory in the scientific sense like, e.g., the theories of relativity, evolution or quantum mechanics. It does not make predictions about the laws of nature. You cannot make statements about reality like: “According to chaos theory, …”, or: “This observation violates chaos theory.”

  Rather, chaos theory is a theory in the mathematical sense like, e.g., set theory or number theory: The word describes a field of study, namely that of chaotic systems or (a little bit wider) non-linear dynamical systems.

• Determinism is the idea that if I completely know the current state of reality, i.e., the position, velocity and other properties of every particle (or whatever reality is actually made of), I can use this information to perfectly predict the future. As measuring the complete state of reality is impossible (even without quantum mechanics), we cannot prove or disprove determinism.

  However, certain theories or models for reality can be deterministic or not. This does not influence their validity per se – how could it if we do not know whether determinism is true or not? We judge models on their capability of correctly describing observations, be they deterministic or not.

• Classical mechanics is a sufficient description of reality for many everyday applications, and for all such applications (i.e., when we use it to model reality), it is deterministic. However, there are some pathological examples for which classical mechanics can be indeterministic if you interpret it in a certain way (for details see this recent question).

• Quantum mechanics is the best known description of reality, whenever gravity isn’t a major factor. For all practical applications (i.e., when we use it to model reality), it is not deterministic. However, there are some ways to interpret quantum mechanics in a way that would make the universe deterministic (hidden variables). This determinism has no bearing on reality though; it is hidden.

  The indeterminism of quantum mechanics is related to the uncertainty principle: Properties of quantum objects (particles) are not fixed, but are drawn from distributions, i.e., they are uncertain. While some property of a particle can be certain (if you just measured it), there is always an uncertain property. The typical example for these properties are the position and velocity of a particle: If you know one, the other is maximally uncertain.

  any given object may, without notice or explanation, spontaneously perform an action it previously was unable to do with a probability that a certain outcome will occur.

  While most of this is technically correct, you probably had something wrong in mind when writing it. Correct is: Objects may do almost everything (e.g., move across the room, vaporize) with a certain probability, but only a small set of behaviours (what we usually observe) has a probability that is measurably different from 0.

• One of the prominent outcomes of chaos theory is the existence of the butterfly effect, i.e., that certain model systems (chaotic systems) can be sensitive to slight perturbations. More specifically, if I take two chaotic model systems that are identical with exception of tiny difference in the initial conditions, their specific behaviour will differ drastically after a while. (Note that their qualitative behaviour will still be comparable.) These chaotic systems most prominently include deterministic ones.

  Now, there are many chaotic models based on classical mechanics that are a good description of reality. Thus even with precision measurements, the behaviour of these systems is only predictable for a very short time. If anything, this unpredictability (and not predictability) was the surprising outcome of chaos theory. Or with other words: Chaos theory taught us that reality is even less predictable than we thought it was.

  Note that if the butterfly effect were the only outcome of chaos theory, it would be rather uninteresting and useless. Its use lies in the capability of understanding other properties of chaotic systems, making qualitative predictions about them and so on.

To summarise, there is no conflict. Chaos theory does not claim that reality is deterministic, and quantum theory does not claim it is random (though that’s arguably the easiest practical approach to reality on atomic scales).

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a future outcome is fully predictable (through rigorous mathematics)

Rigour has little to do with this. Rather, I would use the words tedious, painstaking or Sisyphean.

Answer 2

The uncertainity principle does indicate indeterminism in quantum mechanics, in the sense that uncertainity in position measurement necessitates a corresponding uncertainity in momentum measurement. There is another aspect of indeterminacy in quantum mechanics in the context of measurements, which states that the exact outcome cannot be predicted. On the other hand, Chaos theory dictates the unpredictability of a classical system. There is a key difference between these two words - indeterminism and unpredictability. Indeterministic means what cannot be determined. Eg. In a quantum system, upon measurement, you cannot determine the eigenstate to which it collapses. With reference to Heisenberg's uncertainity, if you know the momentum of a system with some uncertainity, then automatically there will be a bound set on the uncertainity of the position of the system. (I think the word uncertainity in the uncertainity is a rather loose statement creating all kinds of misconceptions, the term Heisenberg's indeterminacy principle is more apt) That is if you take a large number of ensembles of the same state, then you will find that there is always a relation between the uncertainity of position and momentum which is not a characteristic of the measuring apparatus, but fundamentally of nature itself.

The unpredictability in classical systems arises due to chaotic behaviour of the systems. It is however different from the indeterminacy raised in quantum systems. Here the systems are deterministic here (not the case with quantum mechanics), you can exactly predict the future behaviour given the exact initial conditions. However, in practice, you get the initial condition only upto a finite degree of accuracy. This condition coupled with chaotic behaviour gives rise to unpredictability of the system. Note that if the exact initial conditions are known then by the laws of classical mechanics we can get the exact future evolution which is not the case with quantum mechanics, eg, while performing measurement, you don't know the exact future evolution of the system. Unpredictability in classical mechanics is thus more of a problem of determining the initial conditions of the system in question