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In quantum mechanics, the probability, say, that a radioactive atom will decay is well defined. By the Born Rule, it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude.

But if, say, these things were unpredictable and happening for no reason, why does the Born’s rule exist in the first place? Why is it the square of the amplitude and not 1/4 or 1/8?

Secondly, why does the way this work stay constant? If for example, a radioactive atom’s decay point is truly happening for no reason, why doesn’t it just suddenly start decaying with different probabilities? It is easy to imagine this to happen, atleast in a logical sense.

But because this does not happen, does it not still indicate some level of order? If there is order, where does this come from without determinism?

I am aware that “randomness” can create order through the law of large numbers but that originally interpreted randomness as a mere function of ignorance going back to Laplace’s time. This doesn’t fully explain how or why indeterminism can lead to specific probability functions over others and why they generate particular kinds of order over others. If this can’t be explained, how are we so sure that the universe is indeterministic in the first place?

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  • Where did "pure indeterminism" come from? There is a big gap between "pure indeterminism" and fully determined, which is what determinism is. QM is in it, neither fully predictable nor purely random, probabilities just quantify it. One can just as well say that probabilities indicate some level of disorder and ask where it comes from without indeterminism. Why should one of the two extremes be the baseline at all other than being psychologically used to determinism from classical mechanics? And "imagining in a logical sense" borders on oxymoronic. Mathematics "imagines" it in the logical sense
    – Conifold
    Sep 15, 2023 at 9:30
  • @Conifold You accidentally wrote another comment there so you might want to delete the first. Anyhow, the baseline is of course based on intuition and our ordinary experience tells us that the world, atleast in a macroscopic sense, for the most part is ordered. There are many more ways to imagine disordered scenarios than ordered scenarios (even if they result from indeterminism). The more crucial question is why those probabilities in QM don’t change and stay fixed. If things are truly happening for no reason or cause, why is there order at all in the first place? Sep 15, 2023 at 9:35
  • @Conifold I will admit though that either way, one can keep asking the “why?” question. If the universe was deterministic, it would still beg the question of why those laws vs. other laws exist. I am not sure if determinism is apriori more plausible than indeterminism Sep 15, 2023 at 9:37
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    I posted it on both @AgentSmith Sep 15, 2023 at 11:49
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    @thinkingman asks duplicate questions on the physics SE as well, I'd say that was an example of a statistical regularity. Sep 15, 2023 at 15:51

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You seem to be assuming that things are either determined or entirely chaotic. Quantum mechanics is more of a structured hybrid of those two extremes. For example, the energies of electrons in atoms can only take certain values, and there are well-defined probabilities associated with transitions between energy levels. Whether an electron transitions to one of the allowed levels or another does seem, as far as we know, to be entirely down to the luck of the draw.

You can contrast this with a roll of a dice, in that while there is a set of possible results of the roll, and you can assign a probability to each, you could, in principle, work out which result you were going to get if you knew enough about all of the circumstances- eg the angle and speed at which the dice left the hand, the air resistance, the frictional and elastic qualities of the dice and the surface upon which it lands and so on. In that sense, the roll of a dice is determined, and the probabilistic nature of it is really a reflection of our ignorance of all of the underlying details that determine the outcome. By contrast, QM seems to suggest that there is no comparable underlying mechanism with the electron transitions. There are well-established results (see Bell's theorem, for example) which rule out the possibility of ever finding so-called 'hidden variable' theories.

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  • My question is why the wave function the way it is as opposed to something else Sep 15, 2023 at 11:24
  • Good question. Nobody really knows. There are endless arguments about 'what's really going on under the surface' in QM. The wave function itself is a solution to an equation proposed a century ago, which is clearly just a model of what's going on, fudging a lot of the complexities. For example, when you solve the equation for an electron in a solid, you typically adopt the 'one electron approximation' in which the interactions with the trillions of other electrons in the solid are treated as a classical background potential... Sep 15, 2023 at 13:02
  • ... which is clearly not what happens in reality, but nonetheless it gives answers that closely match experimental results. Sep 15, 2023 at 13:03
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When we say Quantum Mechanics is non-deterministic, that has a technical meaning. Specifically, it does not mean that reality is absurd and there are no rules. What is meant by that statement is that the state of a system can, in general, only probabilistically predict the outcome of measurements. Hence, the outcome cannot be predicted or determined with certainty. That is all what "non-determinism" means here, no more, no less.

The universe still follows a very strict set of rules in Quantum Mechanics. For e.g. the future state of a system, in absence of measurements, can be predicted deterministically. And the probabilities of measuring outcomes can be calculated deterministically using the Born rule. This means that, while you cannot predict the outcome of a single measurement, you can make predictions for a repeated set of measurements

But if, say, these things were unpredictable and happening for no reason, why does the Born’s rule exist in the first place?

Again, you have to interpret "non-determinism" in Quantum Mechanics with its specific meaning : the outcome of a single measurement cannot, in general, be predicted. QM is not saying that things are happening for no reason and that there are no rules.

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Things do not happen without a reason. Every event has a cause.

Determinism means that the cause determines the effect with absolute accuracy. There is nothing else affecting the effect, besides the cause.

Reality does not work that way. In reality the cause determines only the probability distribution that the effect will follow. In most cases at quantum scale the probability distribution is the traditional bell curve. The average value has the highest probability. Large numbers of quantum events behave on the average almost as predicted by deterministic laws. Almost, but never quite.

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  • This does not completely explain. When push comes to shove, and nature needs to pick something from the probability curve, how does it do that. As you say, Things do not happen without a reason. Every event has a cause. So what's the reason and cause of that certain probability being 'picked'?
    – Tvde1
    Sep 15, 2023 at 11:35
  • There is no reason why that certain outcome occurs. That is the very point of probabilistic events: They cannot be predicted from previously existing data. The outcome is not knowable before the event occurs. Each probabilistic event creates new information, increase universal complexity and entropy and define the arrow of time. Sep 15, 2023 at 11:56
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You're asking why the math behind reality has certain values and not others. The simple answer is often, if it didn't work out like that, there might not be the possibility for part of existence to ask such questions. Maybe these values are random also and there are other realities where things don't last long enough for life to evolve.

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