Position vs momentum measurement

[hokhani]

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?

 

[Demystifier]

Yes.

 

[Andy Resnick]

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?

I guess it depends on what information you hope to obtain- a point detector does not cover all positions, but then your location measurement is simply 'yes' or 'no' (of course, since you set the position of the detector, you also know the position value). Similarly, a momentum measurement can be likewise performed- a magnet and slit is a momentum filter/selector for charged particles- but your measurement is simply 'yes' or 'no'.

 

[hokhani]

I guess it depends on what information you hope to obtain- a point detector does not cover all positions, but then your location measurement is simply 'yes' or 'no' (of course, since you set the position of the detector, you also know the position value).

It doesn't seem correct. By a position measurement, the particle collapses on one of the position eigenfunctions and in fact you find it there. But, using the point detector, for an extended wave function, almost would never detect the particle.

 

[PeroK]

It doesn't seem correct. By a position measurement, the particle collapses on one of the position eigenfunctions and in fact you find it there. But, using the point detector, for an extended wave function, almost would never detect the particle.

For a position measurement you generally need a detector screen. In the Stern-Gerlach or diffraction experiments, for example.

 

[Andy Resnick]

(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?

 

[jbergman]

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.

 

[hokhani]

(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 $x_0$?