About measuring angular momentum

[snoopies622]

As I understand it, the magnitude of an $L_z$ eigenfunction's value is independent of its argument's $\phi$ coordinate (the longitude). Or, to paraphrase Richard L. Liboff (section 9.3 of Introductory Quantum Mechanics), when a system is in an eigenstate of $L_z, | Y _{l} ^{m} |$ is rotationally symmetric about the z axis.

So then, if I perform an $L_z$ measurement on a particle which is near the origin of some spherical coordinate system and its state changes to an $L_z$ eigenstate, and then I immediately perform a position measurement, the probability of finding it with any particular $\phi$ coordinate (plus or minus whatever $\phi$ angle you choose) is the same.

But what if the particle's original location is very far from the origin, or if it's very massive, or both? Could performing a quantum mechanical observation cause it move all the way to the other side of the origin, for example?

This question seems like it has something to do with the correspondence principle - or perhaps the apparent violation of it - but I cannot quite see it through.

 

[0xDEADBEEF]

Yes the particle jumps to the other side of the coordinate system, but the measurement would normally not succeed. Your measurement always has an "envelope" An area inside which it will give you results. If the particle is far away then the overlap with that envelope is very small, so you will usually not succeed with the measurement.
Furthermore I am not aware of any direct methods to measure Lz around a given axis. Paschen Back type measurements only give you Lz relative to the nucleus, but you do not find out where the nucleus is.

 

[snoopies622]

Well there you go then. I don't know anything about how QM measurements are made.

Incidentally, not long after I submitted this question it occurred to me that one need not even bother with angular momentum to run into this issue. Taking a linear momentum measurement along one axis and then measuring position along that same axis poses the same question. Since a linear momentum eigenfunction extends forever in both directions, one could in theory discover the particle a dozen light years away a moment after measuring it nearby.

Anyway, thanks for the info.

 

[The_Duck]

one could in theory discover the particle a dozen light years away a moment after measuring it nearby.

Note that if you "measure it nearby" then you must be conducting at least a rough position measurement: a measurement of the particle within some experimental apparatus collapses its position space wave function to be entirely within the apparatus. Basically the uncertainty principle ensures that no matter how precise your momentum measurement is, if you then immediately conduct a position measurement you must reobserve the particle within the original momentum-measuring apparatus. So you could only discover the particle "light years away" if your experimental apparatus is light-years in size. But then the question is "light-years away from what?" The original momentum measurement only localized the particle to the volume of the momentum-measuring apparatus. The fact that you then conduct a position measurement and observe the particle at a precise location within this volume doesn't mean the particle has immediately jumped a distance of light-years, as it didn't have a defined position after the momentum measurement.

I think there are issues like you're imagining in a situation like this: measure a particle's position very precisely. Then the particle's momentum uncertainty is very large, so if you conduct another position measurement after a very short time you may find that the particle has traveled a huge distance--perhaps faster than light! The problem here is that the Schrödinger equation doesn't obey special relativity; it's really only an approximation for when relativistic effects are unimportant.

 

[snoopies622]

...if you then immediately conduct a position measurement you must reobserve the particle within the original momentum-measuring apparatus.

Are you saying that if I conduct a position measurement, a momentum measurement, and then another position measurement, I must use the same apparatus for all three measurements?

 

[The_Duck]

I was only talking about two measurements--a "momentum measurement" followed immediately by a "position measurement. But I was trying to say that any real measurement of momentum must also measure position to some extent: unless your experiment is spread out over infinite space the fact that you detected the particle at all means that the particle is somewhere within your equipment. So the wave function collapses into the position of space where the particle could possible have been detected, and then if you conduct an immediate position measurement by any means you must find the particle within that region, since that is where the wave function is nonzero.

Probably this is nitpicking; I'm just trying to point out the differences between ideal measurements and realistic ones.

 

[snoopies622]

...any real measurement of momentum must also measure position to some extent: unless your experiment is spread out over infinite space the fact that you detected the particle at all means that the particle is somewhere within your equipment.

Ah, yes. That makes sense.

Your measurement always has an "envelope" An area inside which it will give you results. If the particle is far away then the overlap with that envelope is very small, so you will usually not succeed with the measurement.

I'd like to learn more about this. So far I've been reading only introductory QM texts which talk all about the expected results of measurements but not the process of measurement itself.