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The measurement of the position of an electron has associated, on the one hand, a quantum uncertainty $\Delta r$ which gives an idea of the probability cloud where the electron can appear, for example it gives the idea of orbital in the case of an electron bound to an atom; and of course $\Delta r$ has nothing to do with our technology. To get information about the probability clouds scientists often work both mathematically and experimentally with a lot of electrons treated statistically (distribution measurements, massive diffraction in crystals, etc).

On the other hand, and that's the topic I want to address, let's suppose the measurement of the position of a single electron, whether bound or free; it has associated a precision or accuracy or error $\delta r$ around the measured position $r$, which is entirely due to the apparatus or to our technological development.

In a former bad-posed question I didn't realize I was confusing these two uncertainties, $\Delta r$ and $\delta r$; thanks to @EmilioPisanty for the comments about that. So I open this new supposedly-well-posed-question.

What is curious is that we always find papers and experiments where they talk a lot about values of measurements, distributions, uncertainties, values of $\Delta r$, clouds, plots of wave functions, shapes of packets, etc, but it's really difficult to find some actual value of the accuracy $\delta r$. How good, approximately, are our experimental devices and technology in 2010-2020 with respect to $\delta r$?

(Also, though this could belong to a new question, any information about the relationship between $\delta r$ and the uncertainty principle, e.g. if this principle is really talking about $\delta r$, $\Delta r$, or both).

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  • $\begingroup$ Suppose you shot a stream of electrons through an adjustable width slit. The uncertainty of the position for passing electrons (when they pass through the slit) would be the slit's width. How thin you could get the slit and still pass electrons through would, of course be related to the wavelength of the electrons, but in principle, the question you're asking depends on the conjugate variables so much that I'm not sure an answer would have any meaning. $\endgroup$ – user121330 Nov 7 '17 at 18:54
  • $\begingroup$ @user121330: Yes, the idea of the electron passing undeviated through a very thin slit (e.g. between two close atoms in a foil) could give a first upper bound of the order of magnitude of this δr (which anyway seems to be an unimportant magnitude, see below). $\endgroup$ – Victor Graus Nov 13 '17 at 16:24
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I can't think of a reason for a high-precision measurement of the location of a low-energy electron in free space; people tend to be more interested in the motion of such electrons, which is easier to measure anyway.

You might like to read about detectors for fast charged particles, such as wire chambers or a silicon strip detectors. In those detectors, a fast electron passing through generates free charges which are collected on the different segments of the detector --- for example, in an wire chamber, most of the ionization is collected on the closest wire, less of the charge on more distant wires, in a repeatable way. Using clever algorithms to analyze the signals from many tracks you can get sub-pixel resolution on the location of the electron's path, but not much information about its location along that path; the time it takes for the electron to cross the detector is typically sub-nanosecond.

When I was tangentially involved an electron-tracking experiment, our detectors with the best theoretical resolution were GEMs. These have the same operating principle as a wire chamber, but the charge amplification happens in very small hole in a copper/kapton surface. I recall those detectors were designed to have sub-millimeter position resolution, but wound up not functioning very well and didn't get much use.

Basically, every PhD student who has built a tracking detector for charged particles will have a section in their dissertation on the position sensitivity of their detector. Since there are ways to access interesting physics even without much position sensitivity, I don't know that you'll find a table published somewhere of "best electron location sensitivities" --- there are too many variables, and the interesting physics is elsewhere.

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  • $\begingroup$ I agree. The initial focus on the HUP's implications on position uncertainty give outsiders and beginners an oversized idea of the importance of the precision of position measurements in modern physics. I can't think of a reason for high-precision measurements of an electron's location in any circumstances, either bound or free. $\endgroup$ – Emilio Pisanty Nov 7 '17 at 19:23
  • $\begingroup$ @rob and everybody: Thanks for the answer and all the comments. I will read about GEMs or similar detectors, and dissertations involving such experiments, which would be a heavy task if I were just looking for a value (often hidden or not present), but after all I will gain a lot of knowledge. I take note that the precision of position is a topic of little importance, the important things being paths in chambers, velocity between detectors, shapes of wave functions, and a lot others. $\endgroup$ – Victor Graus Nov 13 '17 at 16:30

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