TL;DR

Does measurement of momentum like position requires a detector to cover all the momentums?

To measure the position of a quantum particle it seems that the detector must covers all the positions to detect the particle at one point. How about the measurement of momentum? Does this measurement require the momentum detector to cover all the possible values of momentum to show one of the momentum eigenvalues?

hokhani said:

TL;DR: Does measurement of momentum like position requires a detector to cover all the momentums?

To measure the position of a quantum particle it seems that the detector must covers all the positions to detect the particle at one point. How about the measurement of momentum? Does this measurement require the momentum detector to cover all the possible values of momentum to show one of the momentum eigenvalues?

A bit of a misnomer to measure position at one point. Detectors measure a position interval therefore they don't collapse the state to a position eigen-vector but instead just narrow the resulting state vector.

Andy Resnick said:

(Message reply didn't add the quote, Im replying to post #4)

Yes, a point detector would almost never detect the position of a particle. But sometimes it does. Like I said, it depends on what information you are expecting from a measurement.

Surely, you would agree that an array of (independent) point detectors would provide (partial) position information, so what's the minimum number of point detectors that you require?

Suppose that you used a point detector to perhaps detect the particle at ##x_0## and it doesn't detect there. Next you use another detector to detect at ##x_1##. Are you sure that while you are measuring the particle at ##x_1## it would no longer appear at ##x_0##?

Andy Resnick said:

There's no clear answer to this question because you did not specify the relationship between x0 and x1. For example: are they separated by a spacelike or timelike interval?

I'm not sure he meant spacetime positions, just spatial positions--i.e., I think he's implicitly assuming non-relativistic QM.

hokhani said:

TL;DR: Does measurement of momentum like position requires a detector to cover all the momentums?

To measure the position of a quantum particle it seems that the detector must covers all the positions to detect the particle at one point. How about the measurement of momentum? Does this measurement require the momentum detector to cover all the possible values of momentum to show one of the momentum eigenvalues?

In a physical experiment, one can never measure all possible positions or all possible momenta. It is simply not physically possible. However, that is in general not a serious problem. The state is usually prepared in a state that is predominantly located in a finite area and have a momentum spectrum that is also predominantly confined to a finite region. It means that there is a finite but small probability that the photon (assuming quantum optics as the context) is lost. However, since detectors never have perfect detector efficiency, the photon can also be lost even if it falls on the detector. These imperfections contribute to the uncertainty in the measured results, but that is something one always deals with.

PeterDonis said:

I'm not sure he meant spacetime positions, just spatial positions--i.e., I think he's implicitly assuming non-relativistic QM.

Exactly. My discussion is only in the framework of non-relativistic QM.

flippiefanus said:

In a physical experiment, one can never measure all possible positions or all possible momenta...

Right, there are experimental limitations, but I am looking to this subject in a pure theoretical view.