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I'm currently beginning to study QM and came across this interpretation of the expectation value in Griffiths:

Quote: "It emphatically does not mean that if you measure the position of one particle over and over again, $\int x \left|\psi\right|^2dx$ is the average of the results you'll get."

Now,by searching on the web for a little while, I came across this interpretation on the website of the Technical University of Berlin:

Quote: "The result of a single measurement of $ {x}$ can only be predicted to have a certain probability, but if many measurements of the position $ {x}$ under identical conditions are repeated, the average value (expectation value) of $ x$ is $\int x \left|\psi\right|^2dx$".

Maybe I miss some crucial details here and the two interpretations are the same. If they are not, it would be awesome if someone could give an opinion on which I should trust or maybe what I'm missing here.

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  • $\begingroup$ The Technical University of Berlin quote seems to come from this page: www1.itp.tu-berlin.de/brandes/public_html/qm/umist_qm/… If so, then the quote is incorrect: the expression is written differently in the original (in fact, the original contains an equation, not just an expression). $\endgroup$ Nov 20, 2020 at 13:58

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They are not the same.

The first quote is about repeatedly making measurements on the same particle/system, each of which measurements disturbs the state more and more. The mean of the result of this process is not the expectation value.

The second quote is about repeatedly recreating the situation before the first measurement (with a new particle/system if necessary). We do not keep measuring and disturbing a single system after the first measurment. We start with a new undisturbed system each time. This is what is meant by "under identical conditions".

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In most (non-quantum-mechanics) contexts, the second interpretation is correct. Imagine, for example, flipping a coin over and over; since the experiment is repeatable (i.e. the result of doing it once will not impact future trials), you will over time get the expectation value as stated.

However, in quantum mechanics, the experiment is not repeatable because once you perform a measurement once, the wavefunction collapses and you will from then on out get the same result over and over. So you can't expect to get the expected value by repeating the same experiment on the same atom over and over.

I think Griffiths goes on to explain that if you have a very large ensemble of identically prepared atoms, then the expectation value corresponds to the average value from performing the experiment separately on each of these systems. This is the correct way to think about expectation value from the perspective of quantum mechanics.

Hope this helps.

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