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Consider measuring the momentum of an electron. You pass it through some kind of electromagnetic field, it strikes a photodetector (e.g. a CCD), and you back-calculate out the momentum of the particle by how much it curved from a straight-line path. You're taking a position measurement of where the particle hit, and deriving its momentum from this (via a classical equation nonetheless, which seems kind of sketchy to me).

Consider a spin measurement. You use a Stern–Gerlach apparatus and note which way (up or down) the particle curved. Again by looking at the position of where it struck a photodetector.

Consider an energy measurement. You figure out the frequency of an emitted photon by its spectral lines -- once again by measuring the position of these lines relative to some axis.

So am I missing something here? Are there any direct measurements of quantum observables that don't require passing through some "position intermediate"?

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When a gamma ray hits a direct conversion detector, I get a current. I could convert the current into a sound (frequency) and listen. At what point did "position" come into it - other than "the particle was close enough to my sensor to interact"? At no point did a position determination tell me about the energy - yet I have information about the energy. Or are you telling me "the sound resonated with a certain cilia in your inner ear and it was that position that told you what the frequency was"...

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  • $\begingroup$ Isn't the same true of photomultiplier tubes, antennas, Geiger-Müller tubes, etc? It seems like there are lots of measurements that aren't measuring position. $\endgroup$ – Brandon Enright Nov 7 '14 at 17:44
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In a way, you could say it's impossible to make a direct measurement of something without gaining any knowledge about its position. In order to make a direct measurement, you have to directly interact with the thing. Which means you will always learn that at the time of measurement, the object was somewhere within the limits of the measurement apparatus. Even if you have no apparatus and are somehow making an indirect measurement, you will always be able to say that the object you measured is not space-like separated from you, which is still learning information about its position. So if the answer you were looking for was "It's impossible to have zero knowledge about the position of a measured object" then that's what you can have. But if you weren't trying to be so technically correct, then take Floris' answer

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If you want a complement to Floris's example without a moving tympanum or cilia, you could consider an all optical-electronic system. You have some (monochromatic) light whose energy $hν$ you want to measure, a photocathode with a known work function $W$, and some battery cells you can use to bias the gap between the photocathode and anode. You shine the light on the photocathode and you get enough current to drive a light: you know that the energy $hν$ is larger than $W$. Now you add one of your batteries so that there's a voltage $V$ between the cathode and anode, and the light goes off: you know that $hν$ is smaller than $W+V$. Repeat with different $V$ until you have the stopping voltage with the precision that you want.

You're observing the light using the chemistry of your retina, which isn't a position measurement --- you could do the same thing with a one-pixel camera.

The only "position" in this case is that the electrons either do or don't turn around on their way from the cathode to the anode. But that's not a proper position measurement because it depends on the potential difference, not the position of the anode; if I moved the anode around in space but kept the same $V$ across it, I'd still end up with the same stopping potential.

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Yes one can ay everything is eventually a position measurement given an appropriate space where the positions are defined and have the intended meaning.

So if one makes e.g a sound space (indeed there is one in signal processing). All sounds are just positions on this space. And when position changes, the resulting sound changes.

So given such spaces (and the associated intended mapping), indeed one can say all measurements are positions.

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Anything "probabilistic" would not be subject to a "position measurement," because the position is not known!

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