Reasonable measurement of both coordinate and momentum?

[MichPod]

Can a reasonable observable operator be defined which measures a two-component observable, first component for the approximate coordinate and the second for the approximate momentum (so that the precision of each measurement do not contradict Heisenberg inequality)?

I am actually thinking of how to define formally a problem of measuring of the particle speed inside a barrier (for quantum tunneling effect), i.e. can we reasonably ask what is the momentum of the particle inside the barrier? We, of course, can ask what is the average momentum or a momentum distribution, but what about asking what is the momentum IF the particle COULD BE found in some region? Can this sort of problems be reasonably interpreted in the terms of QM?

 

[mfb]

You could make a Fourier transformation of "particle wavefunction in space but limited to the box" and consider this as momentum distribution for the particle in the box. I'm not sure how meaningful that would be, however.

 

[A. Neumaier]

There are POVMs for the simultaneous (and inaccurate) measurement of position and momentum.

 

[bhobba]

There are POVMs for the simultaneous (and inaccurate) measurement of position and momentum.

A POVM is a generalization of the normal measurements you learned about in your QM textbooks. Physically it comes about from using a probe to observe a system then observing the probe, but these days is often taken as the fundamental kind of observation in QM. The important Gleason's theorem, which has a reputation as hard to prove, is much easier using POVM's than the normal Von-Neumann measurements, for example.

See:
http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

Thanks
Bill