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Bohr's correspondence principle and the Born rule are related right?

The correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers. This is due to Born rule working or are they independent?

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  • $\begingroup$ Request for clarification: The correspondence principle is a statement about what quantum mechanics predicts under certain conditions. Can QM predict anything at all without the help of Born's rule? Or are you asking about a more specific connection than this? $\endgroup$ – Chiral Anomaly May 26 at 1:46
  • $\begingroup$ I just want to know how is the correspondence principle relate to the born rule and whether you can have the correspondence principle without any born rule. Im just a layman. $\endgroup$ – Jtl May 26 at 1:52
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    $\begingroup$ Born's rule is always involved in extracting testable predictions from QM, so yes: in cases where QM predicts classical-like behavior (the correspondence principle), it does so with the help of Born's rule. The correspondence principle is basically saying that under the right conditions, classical-like behavior is overwhelmingly more likely (there's Born's rule) than any of the other possibilities. QM's predictions are statistical, and Born's rule is the step that introduces the statistics. That's not specific to the correspondence principle, though, so it might not answer the question. $\endgroup$ – Chiral Anomaly May 26 at 2:00
  • $\begingroup$ Say. Can you apply correspondence principle to a macroscopic object like a baseball? If born rule didn't apply. Would the baseball still look like a baseball? Or do you only use correspondence principle on single system with large quantum numbers and not strictly a macroscopic object? This is because correspondence principle is always tied up to classicality so I wonder if it is about macroscopic object. $\endgroup$ – Jtl May 26 at 2:18
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We really have three questions:

  1. Why is there an inner product (which provides a measure)?

  2. Why is there a probability interpretation?

  3. Given that there is a probability interpretation, why does it have to be given by the measure?

1 is pretty easy to at least construct a plausibility argument for. Quantum mechanics isn't really a physical theory if you omit the inner product. If you don't have an inner product, then you can't define whether an operator is normal. This means you can't define which operators are valid observables.

Without a measure, there will also be difficulties tying QM to the background of space. In the position basis, you can't compare the amplitude or phase of a wavefunction at different points; you can't define differentiation; and you can't define a momentum operator. You can't define what it would mean for a wave packet's envelope to slide along, so you probably can't define galilean invariance or the notion that a hydrogen atom here is the same as a hydrogen atom there. You can't define the Ehrenfest theorem, which along with the inability to recognize valid observables probably means no possibility of making any contact with a classical limit.

3 is fairly straightforward as well. People have formalized it, but there are also easy informal arguments.

As an example of an informal argument, suppose we do double-slit interference with photons. If we run the double-slit experiment for a long enough time, the pattern of dots fills in and becomes very smooth as would have been expected in classical physics. To preserve the correspondence principle, the amount of energy deposited in a given region of the picture over the long run must be proportional to the square of the wave's amplitude. The amount of energy deposited in a certain area depends on the number of photons picked up, which is proportional to the probability of finding any given photon there. This doesn't quite work as a formal proof, because photons don't actually have a wavefunction $\Psi(x)$. For a more careful and rigorous proof, see Gleason's theorem.

The real question is 2. Why should we experience the universe in such a way that when we look back over our experience, we seem to see patterns of behavior that can be summarized by statistical rules? I haven't seen a satisfactory ab initio explanation of this from any more fundamental principles, and if you were looking for such an explanation, I wouldn't know what more fundamental principles to start from.

I think discussions of this sort of thing tend to veer off into religious discussions of MWI versus the Copenhagen interpretation, but IMO that's an orthogonal issue. Neither interpretation pretends to answer #2, and neither interpretation is necessary in order to answer #1 or #3.

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    $\begingroup$ @Jtl: I see. I thought you had already constructed such an argument, and were just looking for confirmation. I'll edit my answer to be more explicit. $\endgroup$ – Ben Crowell May 26 at 0:44
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    $\begingroup$ I think question 1 is more concrete, subtle, and fundamental than your plausibility argument recognizes. If we are to appreciate quantum mechanics as a theory of systems and processes and we are to be so bold as to hope for it to provide to us a quantitative description of nature, then the particular structure of a commutative scalar monoid follows. This review arxiv.org/abs/0905.3010 explains this in detail from section 2.5, the preceding paragraph explaining how the fundamental description was appreciated by Schödinger before the inevitability of the commutivity was appreciated. $\endgroup$ – Diffycue May 26 at 0:49
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    $\begingroup$ @Jtl: It's not a question that admits a yes/no answer. The Born rule doesn't by itself imply the correspondence principle, nor does the correspondence principle by itself imply the Born rule. However, there are logical connections between the two, which I've tried to explain in my answer. if you will google Correspondence Principle and Born Rule. There is only one hit and it is our discussion. Googling combinations of keywords is not a substitute for thinking carefully about difficult topics. I'm also confused by your statement because when I google on this, I get 3850 hits. $\endgroup$ – Ben Crowell May 26 at 1:05
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    $\begingroup$ @Jtl: The harmonic oscillator is a nice example. I like it as a plausibility argument. Self-answers are encouraged on SE, so if you want to do a self-answer based on that example, I'd like to read it. But if you want to argue that the correspondence principle implies the Born rule, solely based on this example, then I expect you to run into trouble constructing a fully rigorous argument. You would need to identify what fundamental principles you would require of a theory, because otherwise there are infinitely many theories that you haven't even considered and therefore can't rule out. $\endgroup$ – Ben Crowell May 26 at 1:28
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    $\begingroup$ @Jtl: OK, an example would be Hawking's proposed modification of the Schrodinger equation using something he called the superscattering operator. This makes QM nonunitary, and therefore we don't have a probability interpretation or a Born rule. But it has problems such as nonconservation of energy. See Banks, Susskind, and Peskin, "Difficulties for the evolution of pure states into mixed states," Nuclear Physics B 244(1984)125. See also arxiv.org/abs/quant-ph/0401062 . $\endgroup$ – Ben Crowell May 26 at 1:48

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