Simultaneity of Uncertainty Principle

[tomwilliam2]

All of the statements of Heisenberg's uncertainty principle that I've read seem to state that there is a fundamental limit on the precision to which you can measure the values of conjugate pairs (like position and momentum) at the same time.

So is this simultaneity necessary? I ask because the course book also says that if you prepare many identical systems in identical states, and perform measurements of position on half of them and momentum on the other half, the results will be consistent with the uncertainty principle. But doesn't this mean you are finding out these values at different times, and therefore not having to sacrifice precision of either one?

If you prepare two identical systems, and test one very precisely for position, then turn to the second system, will there be an inherent limit on the precision with which you can measure momentum? Or will you be able to achieve an arbitrary (but non-zero) level of precision as the measurements were not simultaneous?

Obviously I'm ignoring any technological limitations, as this is not really the point of the question.
Thanks for any help, this one is messing with my mind...

 

[haael]

there is a fundamental limit on the precision to which you can measure the values of conjugate pairs (like position and momentum) at the same time

You have an incorrect interpretation of the mathematical formalism.

First of all, the conjugate operators are noncommuting only at the same time time and space.
[P(x, t), Q(y, s)] = i δ(x - y) δ(t - s)

But it has nothing to do with the time and place where experiments are performed. This relation says that you can not probe two conjugate properties of the same particle, no matter when and where you put your probing device.

Notice first that momentum values of a single particle at different times are not independent. In fact, they are equal if the particle does not interact or they are changing in some predictable way if the particle is in some field. The positions of the particle at different times are also not independent. So, measuring position or momentum at one time gives us some information of these parameters at different times.

You will get the full picture when you imagine the particle as a field extending through the spacetime. The conjugate operators give you the field values at specific time and space points. I.e. the probability that the particle is present at some point.
What we call "particle position" is actually the place where the field has its maximum, or the place where the particle is most likely to be found. This is what you usually want to detect during an experiment.

 

[phinds]

All of the statements of Heisenberg's uncertainty principle that I've read seem to state that there is a fundamental limit on the precision to which you can measure the values of conjugate pairs (like position and momentum) at the same time.

So is this simultaneity necessary? I ask because the course book also says that if you prepare many identical systems in identical states, and perform measurements of position on half of them and momentum on the other half, the results will be consistent with the uncertainty principle. But doesn't this mean you are finding out these values at different times, and therefore not having to sacrifice precision of either one?

If you prepare two identical systems, and test one very precisely for position, then turn to the second system, will there be an inherent limit on the precision with which you can measure momentum? Or will you be able to achieve an arbitrary (but non-zero) level of precision as the measurements were not simultaneous?

Obviously I'm ignoring any technological limitations, as this is not really the point of the question.
Thanks for any help, this one is messing with my mind...

Up until a couple of days ago, I had a similar misunderstanding of the HUP but it was cleared up in a different thread. Basically, the HUP isn't about single measurements, in which you CAN determine both position and momentum to arbitrary precision/accuracy (up to the limits of your measuring device, but in principle not limited), BUT ... what you CANNOT do is perform exactly the same experiment with the exact same setup and get the exact same results. In classical physics, of course, you WOULD expect the same results but quantum measurements are ruled by the HUP.

Every time you measure the position to within a certain accuracy, measurements of the momentum will vary according to the HUP. The more precisely you measure the position, the more variation you will get in the momentum measurements, and this has nothing to do with the limits of measuring devices.

 

[DrChinese]

If you prepare two identical systems, and test one very precisely for position, then turn to the second system, will there be an inherent limit on the precision with which you can measure momentum? Or will you be able to achieve an arbitrary (but non-zero) level of precision as the measurements were not simultaneous?

Obviously I'm ignoring any technological limitations, as this is not really the point of the question.
Thanks for any help, this one is messing with my mind...

Entangled particle pairs represent essentially identical systems (as you are asking for), and yet the HUP applies to this as a single system. This was explored with the EPR paradox. The result was that you can measure position accurately on one and momentum accurately on the other. And yet you still won't know the values of both of these for either particle, as subsequent tests on them will show.