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A friend asked me recently if the Central Limit Theorem holds for quantum systems: i.e., if the distribution of measurements (e.g., of position) for any wavefunction would prove approximately normal, given enough samples.

My gut response was no, because I've encountered plenty of position wavefunctions in introductory QM that wouldn't lead to a normal distribution via Born's rule. But the CLT's requirements seem fairly permissive. What am I missing?

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In the measurement of position the probability of finding a given value is given by one probability distribution. The position is a - in the sense of one - random variable.

The Central Limit Theorem refers to the sum of random variables each of which follows a given probability distribution (in the classical fomrulation of the theorem, the probability distributions of different random variables are taken to be identical, there are formulations in which the probability distributions do not need to be identical).

For more information you can read the Wikipedia article.

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  • $\begingroup$ Thanks—this was the hang-up distinction $\endgroup$
    – Jack Lynch
    Commented Apr 10, 2019 at 19:37
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Let's be clear here: You're asking whether the probability distribution of the normalized sum of many individual quantum measurements of the same quantity necessarily tends to a normal distribution, not whether the probability distribution of the possible outcomes of any single quantum measurement is necessarily a normal distribution. The answer to the latter question is clearly "no". As for the first question, I would say that the answer is "yes". The central limit theorem can be stated as

In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed. Wikipedia: Central Limit Theorem

So whether the individual variables you're averaging result from classical measurements (e.g., the roll of a die) or from quantum measurements (e.g., spin up=1 or spin down=0) and what their probability distributions are is irrelevant to the central limit theorem. If you sum up a lot of individual measurements (either classical or quantum) and normalize them, the outcome will tend to a normal distribution.

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  • $\begingroup$ Are you sure that he is interested in the averaged sum of measurements? I understood that he is interested in the distribution of measurements that comes (for example) from binning different observations. $\endgroup$
    – JTS
    Commented Apr 10, 2019 at 18:29
  • $\begingroup$ I think that he must be interested in the averaged sum of measurements, because otherwise it makes no sense to invoke the Central Limit theorem. A theorem has no relevance unless its conditions are satisfied. $\endgroup$
    – user93237
    Commented Apr 10, 2019 at 19:21
  • $\begingroup$ Ok, let us hear from him: I am not able to interpret his question. $\endgroup$
    – JTS
    Commented Apr 10, 2019 at 19:28
  • $\begingroup$ @JTS's interpretation is correct. I looked elsewhere for definitions of the CLT, but the definition you cited makes the answer abundantly clear. Thanks! $\endgroup$
    – Jack Lynch
    Commented Apr 10, 2019 at 19:36

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