How do we measure a quantum particle's momentum?
The field of spectroscopy is devoted to that issue.
Momentum p is most commonly measured by curvature in a magnetic field B.
There are no unique methods to do this. In http://arxiv.org/abs/cond-mat/0209476" [Broken] (ARPES), the detector has a finite rectangular slit in which one dimension is the energy, while the other is the momentum of the emitted photoelectrons. So you would get a raw image of something like my avatar. For certain types of material (such as 2D, layered material), the transverse momentum of the photoelectrons corresponds to the transverse momentum of such electrons while it is in the material.
Believe it or not, the diffraction pattern that you see on the screen corresponds to the transverse momentum of whatever particle that passed through the single slit.
So do all the methods amount to making a position measurement from which we infer the momentum? How does that square with the uncertainty principle?
Measurement of momentum is always achieved by measurement of position, so I assume that observation of momentum is not how it IS but how it WAS . After measurement of momentum value p, the state cannot keep on |p>. It is a kind of destructive observation.
This is a very important point that IMO is *way* under-emphasized in discussions of QM. We talk rather blithely about measurement of observables in an experimental sense, however the reality is that every measurement we can actually *do* involves either an explicit or implicit measurement of position to one extent or another. The closest thing to a counter-example that I can think of right away might be a direct absorption spectroscopy experiment on a gas confined to a large-volume sample cell. In that case the "uncertainty" in the position of the gas molecules giving rise to the observed spectral lines is fairly large, however you can still say with certainty that they must have been somewhere inside the cell at the time they absorbed a photon(s). So even there there is an implicit measurement of position involved.
At least in my own academic arc, which included a significant amount of formal QM training, this aspect was "pushed under the rug" to the extent that I am not even sure how to address it! I have thought of it on my own several times, and then pushed it back under the rug with a rationalization such as, "well, it must not really be important, or it would be addressed explicitly in QM texts". However, in light of many of the discussions I have participated in here, I think this deserves a closer look. For example, if we look at Zz's answer above, you see that the momentum is inferred from a position measurement of displacement along a given spatial axis. Thus the momentum observable was never really measured at all! Yet we consistently talk as though it had been directly measured. Couldn't this lead to fundamental mis-interpretations of such measurements?
I have to leave now, but I will think about this some more and perhaps start a new thread ....
I would say that neither do position observable.
Separate names with a comma.