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The evolution of the wave function which is determined by Schrödinger’s equation, is said to evolve deterministically.

The wave function represents the probability distribution of potential measurements of a system. Each individual measurement is proposed to be truly random. Thus, it is proposed that the result of the measurement is occurring for no reason.

Now, probability is just related to frequencies. If X has a 5% probability, with enough trials, X will occur 5% of the time.

Now if an electron’s position has a certain probability distribution, shouldn’t there be something, some force, some reason, that is causing that probability distribution to materialize?

If the position at each measurement is truly occurring for no reason, then an amalgamation of things occurring for no reason shouldn’t create a probability distribution. For otherwise, without a guiding physical reason at each measurement that occurs, how would the electron have a “propensity” to be in a position such that over time, the positions agree with the probability distribution defined by the wave function?

An analogy would be coins. In each specific coin toss, we have a reason to prefer no side over another. Given the symmetry of the coin and the constant environmental conditions, we thus have a physical reason that gives each coin toss a “propensity” to land on heads 50% of the time. If the coin was biased, the coin would now have a “propensity” to land on the biased side more often.

Without this propensity which is governed by a physical reason, how could any probability distribution realize?

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    The wave function does not represent the probability distribution, its squared modulus does, and is not related to just frequencies, complex phases also play a physical role as in the Aharonov-Bohm effect. You need to stop equating "for no reason" with "indeterministic", it only confuses everything you write. Those are two different things. "Occurring for no reason shouldn’t create a probability distribution" is an example of a fallacy that stems from that. Analogy to classical probability is faulty as well, QM works differently.
    – Conifold
    Oct 30, 2023 at 19:03
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    Does this answer your question? Is there a reason for why the wave function for a particle is the way it is?
    – Hokon
    Oct 30, 2023 at 20:23
  • @Conifold When you say complex phases play a physical role, you mean they play a role into determining the exact wave function correct? Also, if something was indeterministic and occurred for a reason, wouldn’t that reason imply that it determined it, thus contradicting it being “indeterministic”? In other words, I’m failing to see how something that occurs “indeterministically” can occur for a (sufficient, atleast) reason.
    – user62907
    Oct 30, 2023 at 20:32
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    We've been over this multiple times. Indeterministic just means insufficient reasons, it means neither that they determine something nor that that there is "no reason". And, unlike in the classical case, quantum lattice of events is non-distributive, so you cannot split indeterministic events into determined and random parts, they are entangled. You are running in circles with your classical analogies that essentially presuppose classical hidden variables for "reasons".
    – Conifold
    Oct 30, 2023 at 23:08
  • The reason for the result of the measurement is the combination of how the the state under observation was prepared and how the device or mechanism used to produce the measurement was prepared.
    – Dave
    Nov 1, 2023 at 22:29

4 Answers 4

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Quantum mechanics did not find any reason for the result of measuring a given single event from microphysics. And the Copenhagen interpretation argues that there is no reason.

This result was disturbing already from the beginning in the 1920's.

Lessons learned: We cannot prescribe nature the rules that we prefer or are used to from mesocosmos. Instead we have to accommodate our expectations to the answer, nature gives to our questions, the experiments.

Keywords: Copenhagen interpretation, hidden parameters in this blog.

Aside: The Schroedinger equation is a differential equation not for the probability distribution but for a function psi which attains values in the complex numbers. But the modulus | psi | ^2, a real number, is the probability distribution.

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Your question gets to the heart of the core philosophical challenge of quantum mechanics, which is how do we interpret the mathematics of quantum mechanics - what part of nature is it (specifically) describing when we make a "measurement" from a viewpoint of scientific realism. From this comes the myriad of interpretations of quantum mechanics, each with a different answer to your question. The first two paragraphs of the SEP: Quantum Mechanics speak to exactly this issue:

Quantum mechanics is, at least at first glance and at least in part, a mathematical machine for predicting the behaviors of microscopic particles — or, at least, of the measuring instruments we use to explore those behaviors — and in that capacity, it is spectacularly successful: in terms of power and precision, head and shoulders above any theory we have ever had. Mathematically, the theory is well understood; we know what its parts are, how they are put together, and why, in the mechanical sense (i.e., in a sense that can be answered by describing the internal grinding of gear against gear), the whole thing performs the way it does, how the information that gets fed in at one end is converted into what comes out the other. The question of what kind of a world it describes, however, is controversial; there is very little agreement, among physicists and among philosophers, about what the world is like according to quantum mechanics. Minimally interpreted, the theory describes a set of facts about the way the microscopic world impinges on the macroscopic one, how it affects our measuring instruments, described in everyday language or the language of classical mechanics. Disagreement centers on the question of what a microscopic world, which affects our apparatuses in the prescribed manner, is, or even could be, like intrinsically; or how those apparatuses could themselves be built out of microscopic parts of the sort the theory describes.1

That is what an interpretation of the theory would provide: a proper account of what the world is like according to quantum mechanics, intrinsically and from the bottom up. The problems with giving an interpretation (not just a comforting, homey sort of interpretation, i.e., not just an interpretation according to which the world isn’t too different from the familiar world of common sense, but any interpretation at all) are dealt with in other sections of this encyclopedia. Here, we are concerned only with the mathematical heart of the theory, the theory in its capacity as a mathematical machine, and — whatever is true of the rest of it — this part of the theory makes exquisitely good sense.

(Source: Quantum Mechanics, Stanford Encyclopedia of Philosophy, accessed 1:52 US Central Time, https://plato.stanford.edu/entries/qm/)

Even more challenging is that we can't necessarily take the wave function itself as indicative of anything, since there is not one but at least nine equivalent formulations!

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    +1 for pointing to the paper by Styer et al.
    – Jo Wehler
    Oct 30, 2023 at 19:47
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I'm not sure if this observation will help, but you can easily imagine arrangements in which randomness leads to clear-cut probabilities. For example, suppose I have a large tray which I divide into three areas, A,B and C, whose relative sizes are 1,2,and 3. If I leave the tray on the ground outside in the rain, I can still say that the chance of a raindrop landing in area C is 50%, even though the rain is randomly distributed. The probabilities arise from a combination of a random set of events (ie the rain-drops), plus a structure (ie the relative sizes of the subdivisions in the tray. There is no reason why a particular rain-drop landed where it did.

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  • This to me is no different from the dice example. There is a physical reason, C being the largest, which gives each rain drop a “propensity” to land on C over A for example. I see no such thing in quantum mechanics.
    – user62907
    Oct 30, 2023 at 22:12
  • @thinkingman, well I did say "I'm not sure if this observation will help" so feel free to downvote. I would do it myself but SE won't let me! Maybe you could find something analogous in QM in the following way. Suppose an electron, say, has a wave function with discrete Eigen-functions for some property such as its energy. The eigenfunctions are like the partitioned sections in the tray- if you measure the energy of the electron, its wave-function has to end up as one of the eigenfunctions, like the raindrop has to end up in one of the smaller areas. Oct 31, 2023 at 6:12
  • I can just imagine the storm of downvotes that would provoke on Physics SE, so please don't take it too seriously, but just as a hand wavy argument, you might say that the wave-function is triggered to change by some random factor, and once triggered it must end up in one of the possible Eigen-states. There is definitely a nice analogy between the relative areas of the sub-divisions in the tray and the fact that in the blend of Eigen-states that form a given wave-function, each Eigen-state is a numerical portion of the mix. Just as a raindrop is more likely to... Oct 31, 2023 at 6:19
  • ...land in the largest sub-area of the tray, an electron's wave function after a randomly-triggered jump is most likely to end up in the eigenstate that is most present in the blend. Oct 31, 2023 at 6:21
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You're basically rehashing some past debates, which makes this easy in my opinion. At the most general level...in the 1950's we got (a version of) Reichenbach's Principle: every statistical correlation entails specific causes. I see no barrier in your text to equating correlations and probability distributions. We can clearly see there are correlations in QM statistical data, and hence must seemingly conclude there are causal reasons behind the statistics. The correlations in QM include different probabilities at parts of the wavefunction (which depends on space and time), and when pairs of particles have predictable joint outcomes. Reichenbach's principle is larger than the debate about QM, it's about probabilities and causes in general, although some have directly applied it to QM (in favor of there being hidden, even local, causes/variables).

But QM also includes theories like GRWf, where there are theoretically posited random physical processes. Is that a reason or not, seems like not. Are there any non-tautological reasons in many worlds if it's all just self-locating? Again seems like not.

So it would seem the gamut varies from yes to no. GRWf is truly random, there is no causal model. Without a causal model existing, there are no reasons and R's principle is wrong. Measurement outcomes would not have non-probabilistic reasons, contra R's principle.

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