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I have read these questions:

Is the wavefunction a real physical wave or only a mathematical abstraction?

The wavefunction gives you the probability description of a particle's position (and other characteristics).

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi, respectively).

In Born's statistical interpretation in non-relativistic quantum mechanics,[8][9][10] the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle's being detected at a given place – or having a given momentum – at a given time, and possibly having definite values for discrete degrees of freedom.

But the wavefunction is just information about the particle. But where is this information stored?

Is it stored in the fabric of spacetime?

Or is it stored in the particle somehow that we do not see?

Or is it stored in an extra dimension that we do not understand?

Question:

  1. Where is the information of the wavefunction stored?

After the comments, here is another question, assuming that the question where it is stored, can be understood as what the possible range of values for probabilities are:

  1. Is it OK to say that the state of a quantum particle is stored in a projectivized Hilbert space?
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Árpád Szendrei
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    In classical mechanics, where is the information about a particle's position stored? I ask because it's not clear (at least to me) what "stored" means here, and your answer to this question might clear that up. – WillO Aug 12 '18 at 17:52
  • @WillO in classical mechanics, the particle is not a quantum object. At a certain point in time, the particle has a 100% probability to be at a certain position, and 0 to be anywhere else. There is no information (at a certain point in time) stored about other places in space. In QM, the wavefunction stores much more info. It stores info about all points in space. The particle (at a certain point in time) might be at a certain point in space with a less then 100% probability, and at the same time with a certain probability at another place. – Árpád Szendrei Aug 12 '18 at 17:58
  • At the same time is important, because that is what is different from classical. QM wavefunctions store info about all of space for a certain particle. Of course you could say that in classical all that info is there too, is just that info is all zero for all other places in space (for a give point in time). This is the point. Storing zero only makes sense if zero could change. In classical, those zeros cannot change ever. So they need not be stored. Classical is just a description. For Micro object it does not work as good as QM. QM works better, so it must be closer to reality. – Árpád Szendrei Aug 12 '18 at 18:00
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    And in classical mechanics, where is the information about that certain position stored? – WillO Aug 12 '18 at 18:02
  • And if QM is closer to reality, then that info, the non zero probabilities for all space (at a certain point in time) need to be stored somewhere physically. I am asking where is that stored, is there any theory on that? I have found nothing. – Árpád Szendrei Aug 12 '18 at 18:02
  • "in classical those zeros could not change ever"---so in classical mechanics, no particle ever moves? – WillO Aug 12 '18 at 18:06
  • @WillO I understand your point. I am saying that QM is a better description for micro objects. In classical what you are referring to is the position of the particle, which is the particle itself in spacetime. In classical the particle's physical existence could create that info. Though, based on what you are saying, spacetime (vacuum) stores the info that the probability of the particle (or better a macro object) is not being anywhere else (assuming there would be only one object in vacuum to make it simple). In QM it is not like that. – Árpád Szendrei Aug 12 '18 at 18:08
  • @Willo At a certain point in time. That is not motion. That is just a static state frozen in the time dimension. – Árpád Szendrei Aug 12 '18 at 18:08
  • I understand what you are saying. In classical, what you say works for macro objects, even for particles. But that is not how reality works in the micro level. QM is better at the micro level, and those probabilities cannot be just presented by the physical particle itself. Though you are right, that motion will make that info even more necessary to be stored somewhere, like momentum. The four vectors and the probabilities in the wavefunction need to be stored somewhere. – Árpád Szendrei Aug 12 '18 at 18:11
  • I believe you are saying that it is all stored in spacetime and the physical state of the particle. That is OK for classical. – Árpád Szendrei Aug 12 '18 at 18:12
  • What you are saying, would lead to an experiment, where we could detect the same particle at the same time at different places. That is not how it works in QM. These are just probabilities. We have to repeat the experiment. But we can never find the particle at the same time at different places. That info about the probabilities must be stored somewhere. – Árpád Szendrei Aug 12 '18 at 18:14
  • This might be a good question for SE.QuantumComputing. – Nat Aug 12 '18 at 18:40
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    Okay---So I understand you to be saying that at a point in time, the position of a classical particle is "stored" in (presumably 3-dimensional) space. So apparently when you ask "where is it stored?" you mean to ask "what is its range of possible values?". (If you don't mean this, I wish you would just straightforwardly tell us what you do mean.) The answer, then, is that the state of a quantum particle is "stored" in a projectivized Hilbert space. – WillO Aug 12 '18 at 20:19
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    take the moon. It is in an (x,y,z,t) point with respect to the earth. Where is the information that it has a specific calculable orbit stored? This line of thinking leads to platonic ideals : mathematics creates reality.(contrasted to reality is described by mathematics). – anna v Aug 13 '18 at 05:39
  • @annav I understand and agree that math only describes reality. With macro objects and gravity (spacetime itself is bent) it is understandable. What I do not understand, is how micro objects in QM can be like that. the wavefunction should be like a map that describes the particle's positions' probabilities for all of space (for one moment in time). It is hard to understand that a particle could be at the same time at different places, and that is reality and the wavefunction just describes that? – Árpád Szendrei Aug 13 '18 at 06:04
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    <Where is the information of the wavefunction stored?> I think any 'information' about any state of a particle is stored in the observer's measurement data file. – drvrm Aug 13 '18 at 09:20
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    What is understandable? Classical solutions are stored in our books and our computers. the same is true for the probability functions describing elementary particles. the two are on par. Your question turns into questioning "why not deterministic" and the answer is "because the mathematics fitting data needs i probabilistic formulation" – anna v Aug 13 '18 at 12:21

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Concepts of wave function and measurement are quite interrelated. You apply external macroscopic measuring device towards a microscopic object, then you observe how the state of this macrospic device changed. From this change you infer what was a state of microscopic quantum object. Please see Basic Concepts of Quantum Mechanics, Chapter 1, by L&L.

The state of microscopic object can be described, as well as its future behaviour can predicted with a high degree of precision by introduction of the abstraction called wave function. So the wave function, namely its parameters, itself defines microscopic state. Hence information about its parameters infered through macroscopic measurement is stored in the observer's dataset.

Hence:

  1. Q: Where is the information of the wavefunction stored?

    • A: In the observer's dataset.

  2. Is it OK to say that the state of a quantum particle is stored in a projectivized Hilbert space?

    • A: No, see the item above.
Artem
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  • I think the information is in the particle itself as it goes for mass, charges and more. Also in the surrounding particles tru the interactions (Earth gravitationaly knowns where the Moon is). – Mercury Jan 04 '19 at 20:46