# Position measurement in Non-Relativistic QM

I've read several times in textbooks that in NR-QM you can measure with exact precision the position of a particle if you don't care at all for the momentum (because the uncertainty of the momentum will be infinite), it always seemed reasonable enough for me but now that I think about it, it should be impossible in principle because the wave function would collapse to a delta function, and the delta function is not in the hilbert space of the wave functions.

Any thoughts?

## Answers and Replies

atyy
Science Advisor
The quick answer is that one doesn't need collapse unless one is doing another measurement after the first. So we can just declare the measurement done, and the particle destroyed after that.

However, the collapse rule is actually more general, and doesn't have to be a projection onto an eigenvector of the observable. The heuristic way to see this is that what you consider a measurement is subjective, so you can collapse it onto the eigenvector and rotate it, and call that process the measurement.

OK, but that still doesn't help with continuous variables, since the eigenvector itself is not a legitimate wave function. For continuous variables the collapse is defined by Eq 3 of http://arxiv.org/abs/0706.3526. Remarkably, Ozawa discovered that there is a measurement model that will give a sharp position measurement, which is summarized in section 2.3.2 of that paper.

Last edited:
• Greg Bernhardt
Demystifier
Science Advisor
Gold Member
I've read several times in textbooks that in NR-QM you can measure with exact precision the position of a particle if you don't care at all for the momentum (because the uncertainty of the momentum will be infinite), it always seemed reasonable enough for me but now that I think about it, it should be impossible in principle because the wave function would collapse to a delta function, and the delta function is not in the hilbert space of the wave functions.

Any thoughts?
That means that you cannot measure position with infinite precision, but you can measure position with an arbitrarily good finite precision. For all practical purposes, this is the same.

• bhobba
bhobba
Mentor
That means that you cannot measure position with infinite precision, but you can measure position with an arbitrarily good finite precision. For all practical purposes, this is the same.

This is seen by the fact a state with perfect position is a Dirac Delta Function - which isn't really a function, and certainly doesn't belong to a Hilbert space.

Thanks
Bill