DaleSpam said:
I thought that it was more that you could not prepare a state where both were well defined, but you could measure them as accurately as you like.
phinds said:
Yes, that's my understanding as well, but there are strong proponents here on this forum for both points of view --- (1) you can measure both simultaneously to any degree of accuracy but the next measurement from the same setup won't give the same values, and (2) setup doesn't matter since you can't measure them simultaneously to arbitrary accuracy anyway. I tried once to get it pinned down but the thread I started didn't achieve that, just made it clear that there ARE two points of view here on this forum, so I've been unable to tell from this forum which it is.
ZapperZ and I had a discussion about this, and I believe there is only one point of view (I can't find the post now) - non-commuting observables cannot be simultaneously and accurately measured on an arbitrary unknown state. People often read ZapperZ's blog post as meaning that simultaneously accurate measurement of position and momentum is possible, but that is not what he means. What he means is that in a double slit experiment, if one puts the screen far away, and measures position accurately at the screen, then one can use that position result to accurately calculate (the "simple" method he uses gives the right answer, but one can also obtain it by a correct quantum mechanical calculation using the Fraunhofer limit), and thus to accurately measure the transverse momentum of the particle immediately after the slit. Thus a later accurate position measurement can be used to accurately measure an earlier, non-simultaneous, momentum. As the slit is narrowed, the initial position does become more certain, but the narrowing of the slit changes the wave function immediately after the slit, so it is not a more accurate measurement of "the same" wave function.
So to summarize:
(1) The textbook uncertainty relations do not refer to simultaneous measurement. They say that a state cannot have simultaneously well-defined position and momentum. This means that in accurate measurements of position on an ensemble of particles in a state and separate, non-simultaneous accurate measurements of momentum on a different ensemble of particles in the same state will yield position and momentum spreads that satisfy the textbook uncertainty relation.
(2) The textbook uncertainty relations are derived from non-commuting observables. It is not possible to simultaneously and accurately measure non-commuting observables on an arbitrary unknown state. There are exceptions if the state is not arbitrary and we know something about it. For example, if the state is known completely, we can simply write down simultaneous, accurate "measurement results". Also, if we know (for discrete variables) that the state is one of several eigenstates of an observable, then a measurement of that observable will leave the state undisturbed, so that the same state is still available for a non-commuting observable to be measured. Another famous special state is the EPR state that allows simultaneous measurement of position and momentum because of entanglement. (There's a "controversy" in the literature, but it's semantics about how one should define the "measurement error". Let's just say that for the various choices of "measurement error", I believe these papers are correct.)
http://arxiv.org/abs/1306.1565
Proof of Heisenberg's error-disturbance relation
Paul Busch, Pekka Lahti, Reinhard F. Werner
http://arxiv.org/abs/1304.2071
How well can one jointly measure two incompatible observables on a given quantum state?
Cyril Branciard
http://arxiv.org/abs/1212.2815
Correlations between detectors allow violation of the Heisenberg noise-disturbance principle for position and momentum measurements
Antonio Di Lorenzo
There are a couple of famous, earlier papers in the literature. Park and Margenau's 1968 paper is correct, because it shows the simultaneous measurement of conjugate position and momentum for a restricted, non-arbitary set of states. Ballentine's 1970 review is wrong, because position and momentum are not conjugate in his example.