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Let $A$ be an observable, then, is there a way to experimentally determine the mean of $\hat A$, namely $\langle \hat A \rangle $?

I mean, for example, consider the position operator; it is intuitively plausible that if a particle can be in $x =0$ (for a fixed reference frame), then another particle can also be in $x = 10, 100, 1000$, or for any other value, there is nothing special about $x=0$, or another value. However, we never went to the edge of our galaxy and tested whether a particle can indeed exists there.

(Of course astronomical observations can be made for lots regions in space, I'm sure there are some regions in space where nobody has ever observed another matter there; Addition to that, even we observed any matter, we still wouldn't know whether is made of matter or antimatter - there might be some update on that topic which I'm now aware of the I'm digressing).

This is, in a sense, a trivial example, but, in general, how can we measure $\langle \hat A \rangle$ ?

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When you want to measure the expectation value (i.e. "mean") of an operator corresponding to a particular observable, applied to a given state, you generally do the following:

1) Produce or obtain a bunch of identical copies of the state. If they can't be made exactly identical, then this adds to your systematic uncertainties (one way to account for possible contamination is to assume you have a mixed state rather than a pure state at the outset).

2) Measure the observable on each copy, correcting for any bias and accounting for any systematic uncertainty introduced by the particular measurement process.

3) Take the average of those measurements, accounting for the statistical uncertainty produced by a finite sample size.

Now, to address your particular example:

for example, consider the position operator; it is intuitively plausible that if a particle can be in x=0 (for a fixed reference frame), then if should also be in x=10,100,1000, or for any other value.

Not necessarily. The probability of a particular particle being at a particular position is determined by the wavefunction of that particle. If our particle's wavefunction reflects its being confined in, for example, a very strong potential well, then its probability of being outside the well is so vanishingly low that we will probably never measure it outside the well, ever. As such, the mean value of the position is barely affected at all from the contributions outside the well. In the interest of being thorough, the uncertainty in the expectation value of position due to the effect of tunneling outside this well can be accounted for as part of the systematic uncertainties I mentioned earlier. Similar additions to the uncertainty can be made for wavefunctions subject to other potentials. Even completely unconfined wavefunctions must still be normalizable, and so cannot have infinite extent.

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  • $\begingroup$ What I meant with in that argument was that, some matter or particle can exists at $x= c$ for any $c \in \mathbb{R}$, so for any $x \in \mathbb{R}$ is a possible outcome of $\bar x$; of course, I'm not considering any bounded case in here; what I'm simply trying to convey is that there is no such case where for some $c \in \mathbb{R}$ s.t at $x = c$, there is no possibility that a particle can exists at $x = c$. $\endgroup$ – onurcanbektas Jan 22 at 4:26
  • $\begingroup$ How does assuming that we have a mixed state accounts for the non-identicalness of the systems ? $\endgroup$ – onurcanbektas Jan 22 at 4:58
  • $\begingroup$ For example, if some particles are in $x_1$ and some $x_2$ state, then should I say that all the particles are in $x_0 := N(x_1 + x_2)$ state ? This wouldn't represent any of the particles in our sample, so I'm not sure what you meant. $\endgroup$ – onurcanbektas Jan 22 at 5:02
  • $\begingroup$ @onurcanbektas For a particle in the $n=1$ energy eigenstate of a finite square well potential, the probability of finding the particle at $x=0$ is zero. $\endgroup$ – probably_someone Jan 22 at 9:59
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    $\begingroup$ @onurcanbektas As an example of its usage, suppose you wanted to make a bunch of states corresponding to electrons which are spin-up when measured along the $z$-axis. Any process you use to do this is going to be imperfect, and there is a small (classical) probability that your process will create some other state. For example, there might be a 99% chance that the state you produce is $|\uparrow\rangle$ and a 1% chance that it's the nearby state $\sqrt{0.9}|\uparrow\rangle+\sqrt{0.1}|\downarrow\rangle$. This is a mixed state; there's a lot of theory devoted to dealing with them. $\endgroup$ – probably_someone Jan 22 at 10:13

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