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How do we know that the observed location of a electron (or any quantum object) is purely random (there is no way to predict it) within the probability-function instead of normal randomness (we don't know how to predict it)?

As an analogy if someone would only be able to measure the amount of people (amount of electrons) and their individual IQ (location) he would see a perfect distribution (obviously only if he has enough samples) within a specific range (probability) and each individual measurement seems purely random (QM-randomness). He would then assume that their IQ is purely random as he has just not enough knowledge to prove otherwise (he hasn't spoken to any of them, hasn't analysed any in respect to education nor has he analysed their DNA).

Could the same be happening when we observe electrons or have we proven that their location is purely random?

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    $\begingroup$ Possible duplicate of How do we know that certain quantum effects are random? $\endgroup$
    – Stéphane Rollandin
    Commented Jun 4, 2019 at 12:23
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    $\begingroup$ Is this an empirical question? Or is this a question about quantum theory? If it's an empirical question, then what does "random" mean empirically? The usual meaning is "we don't know how to predict it, except for the distribution," but then the question is asking "How do we know we don't know how to predict it?" $\endgroup$
    – Chiral Anomaly
    Commented Jun 4, 2019 at 12:41
  • $\begingroup$ @ChiralAnomaly Thanks for your response as this is very fundamental to my question. With random I mean "There is no way to predict it" and not "We don't know how to predict it". If it's random the way you define it, my question would be answered. My question arised as I've always been told that in quantum mechanics there is this "true randomness" with nothing causing it. I want to know if this truly is a "true randomness" and if we have proven so. $\endgroup$
    – LeStuder
    Commented Jun 4, 2019 at 12:47
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    $\begingroup$ @Leander BTW, those things like education & DNA correspond to local hidden properties of the electron which determine its location but which we are unable to measure for some reason. But the Bell inequalities tell us that the electron doesn't actually have any such local hidden properties, because there is no way to assign values to those properties which is mathematically consistent with the observed results in Bell-type experiments. $\endgroup$
    – PM 2Ring
    Commented Jun 5, 2019 at 9:31
  • $\begingroup$ @PM2Ring Thanks a lot for you comment, you pointed me into the right direction. So we are only certain of the fact that there are no local hidden variables, but not non-local hidden variables? If you write your explanation as an answer I'll accept it. Maybe mention that it's not possible for local hidden variables to exist while the possibility for non-local hidden variables is still there. $\endgroup$
    – LeStuder
    Commented Jun 5, 2019 at 14:52

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How do we know that the observed location of a electron (or any quantum object) is purely random within the probability-function?

The location of an electron is not random by any means, rather it is exactly determined by the coefficients of its expansion onto a basis. Namely let the state of the electron be described by $$ |\psi\rangle = \sum_n c_n |a_n\rangle $$ where $|a_n\rangle$ is a complete basis of the Hilbert space and eigenstates of an observable $\hat{A}$. This means that if we perform infinite measurements of the physical quantity associated to $\hat{A}$, we found the value $a_n$ exactly $|c_n|^2$ times each. Replace $\hat{A}$ with the position operator and the coefficients will represent the probability of finding the particle in that position if we make infinite measurements.

The fact that you get different values if you perform many different measurements of the same quantity is not due to the fact that you do not have enough information, rather it is a definition in quantum mechanics. Any observable can have in principle infinitely many different values when measured on a state (unless you are on an eigenstate of said observable).

The above assumptions have been proven right experimentally, in particular the position of an electron according to the above was one of the first precise proofs of quantum mechanics, see double-slit experiment.

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  • $\begingroup$ If I understand right you say that there is a true randomness (There is no way to predict it) rather than a "normal" randomness (We don't know how to predict it else than that infinite measurements form the probability-function) and that this is a definition in QM? Isn't that a invalid argument? QM itself has some ways which prevent us from knowing more (measurement problem) and thus we need to assume that we don't know all information. To solve this problem physicists make it a fundamental definition that there is true ramdomness, knowingly that they can't know if it wasn't. $\endgroup$
    – LeStuder
    Commented Jun 5, 2019 at 6:49
  • $\begingroup$ @LeanderStuder To be honest I don't understand your comment. I argued that there is nothing really random but everything is perfectly predicted by the coefficients of the state expansion; notice that different values does NOT mean random. Also, I don't really understand the downvote either (anyone care to explain?). $\endgroup$
    – gented
    Commented Jun 5, 2019 at 7:56
  • $\begingroup$ I think PM2Ring in the comment-section of the initial question has answered my question in appropriate language even if I still need to understand how exactly the bell-inequalities prevent us from assigning local hidden properties to electrons. I guess that the downvote, which is not from me, is because you have answered a layman question with a overly technical jargon which is obviously not understandable by me (not your answer nor your comment) even if it is technically correct. $\endgroup$
    – LeStuder
    Commented Jun 5, 2019 at 14:57

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