How is anything *not* ultimately a position measurement?

Question

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"?

Answer 1

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"...

Answer 2

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

Answer 3

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.

Answer 4

In classical physics all objects have a position (or they are extended) in space. In that sense every measurement is however associated to a location in space. A more delicate situation takes place concerning quantum systems where, for instance particles, have no precise position in space. However, the (quantum) measurement apparatus of anything always occupies a position in space. In that sense we can say that every realistic quantum measurement procedure provides also some spatial location for the system and its post measurement state. The notion of quantum instrument based on POVMs (and Kraus decomposition) instead of PVMs, selfadjoint operators and the Luders-von Neumann projection postulate is able to include this spatial information. There is a huge literature on the subject. I can suggest the last edition of the book by Busch and coworkers on quantum measurement.

Answer 5

Yes one can say 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.

Answer 6

Fundamentally, position and time intervals are the ultimate geometrical measurements. But we also have to take into account statistics: events counting. This would include other kind of measurements in physics which cannot be reduced to position/time measurements. Personally, I strongly believe that all measurements can ultimately be reduced to these two classes of measurements and information acquisition.

Answer 7

You're close, but you still have a misconception. We don't actually measure the position of anything, because position is an abstract concept that would require infinite precision to measure. What we actually measure is counts or events. It just so happens that locality in spacetime means that two events are distinguishable when they occur at different locations. Consider the examples raised so far:

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.

What you measured isn't the location of the particle, you measured "1 electron hit this element of the CCD". You can infer much about the position from what you know of the structure of the CCD, but the pixels have finite sizes.

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.

Not likely. I did Stern–Gerlach in an undergrad lab, and we used a movable wire shielded and watched how the current changed as a function of position, iirc. In other words, we measured current, fundamentally counting the rate at which electrons passed through a circuit element, and couldn't localize the atoms better than the width of the wire.

In the old days, they just accumulated the metal on a glass plate, and looked at how the opacity changed.

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.

I think you mean after dispersing the photon off of a grating or through a prism, right? Same criticism: you count photons and infer position. You could also measure the energy using calorimetry: you shine many examples of the photons on a sensitive thermometer and see how much the temperature changes. On old school thermometers that means measuring how much the surface of some fluid, like mercury, changed. In more modern thermometers (e.g. thermistors and thermocouples) this becomes a difference in voltage or current. Both of those measurements are done by counting/timing these days.

In fact, the only tricky examples I can think of are situations like radio and static electric fields, where the photons are coherent. There it doesn't even make sense to talk about observing 1 photon, or even a large number of discrete ones - you observe a property of the whole bunch. Even so, you can think of it as photons without too much trouble in most cases.

In fact, the insight that we observe counts is at the core of second quantization. $\mathbf{x}$ gets demoted from an observable to a parameter, to match $t$, and the field strengths become operators. It's my understanding that the field strength operators are not directly observable, but observables are built from them. Long story short, the actual observables are all built from integrating number density operators over some acceptance range of a detector.

Answer 8

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