Is Uncertainty intrinsic property or a consequence of measurement ?

[bhobba]

I think in this regard its also important to understand exactly the axiomatic basis of QM, as well as a way of looking at them that brings out clearly what they are saying. I will base the following on the two axioms in Ballentine - QM - A Modern development. Its a very interesting fact that it really rests on just two axioms. There is a bit more to it, but they are more or less along the lines of reasonableness assumptions such as the probability of outcomes should not dependent on coordinate systems ie symmetry.

Imagine we have a system and some observational apparatus that has n possible outcomes associated with values yi. Note we are assuming such observational apparatus exists. It is a fact they do. Their accuracy and precision is a matter for experimental guys to investigate - it is simply assumed they exist and have some accuracy and precision.

This immediately suggests a vector and to bring this out I will write it as Ʃ yi |bi>. Now we have a problem - the |bi> are freely chosen - they are simply man made things that follow from a theorem on vector spaces - fundamental physics can not depend on that. To get around it QM replaces the |bi> by |bi><bi| to give the operator Ʃ yi |bi><bi| - which is basis independent. In this way observations are associated with Hermitian operators. This is the first axiom in Ballentine, and heuristically why its reasonable.

Next we have this wonderful theorem Gleason's theorem that I alluded to previously, which, basically, follows from the above axiom:
http://kof.physto.se/theses/helena-master.pdf

This is the second axioms in Ballentine's treatment.

This means a state is simply a mathematical requirement to allow us to calculate expected values in QM. It may or may not be real - there is no way to tell. Its very similar to the role probabilities play in probability theory. In fact in QM you can also calculate probabilities. Most people would say probabilities don't exist in a real sense and why I personally don't think the state is real - but that's just my view - as far as we can tell today its an open question.

Basically QM is a theory about the probabilities of outcomes of observation, if we were to observe it. The fundamental assumption is devices capable of observing quantum systems exist, and they will give an actual outcome. There accuracy and precision is a characteristic of the device and its up experimentation to determine what that is.

Thanks
Bill

 

[bhobba]

So what is so different in QM statistics and classical statistics, that former makes nature itself probabilistic and later is used because it is the only simple way to handle large number of degrees of freedom but considers nature to be non-probabilistic.

Well if you read the paper I linked to on QM from 5 reasonable axioms you see at a mathematical level QM allows a continuous transformation between so called pure states. Classical probability doesn't do that. That is the key difference mathematically.

In fact further work has been done that shows entanglement is the fundamental difference:
http://arxiv.org/abs/0911.0695

Basically only two reasonable theories exist to model physical systems as generalized probability models - standard probability theory and QM. But of those two QM is the only one that allows entanglement, which has been experimentally demonstrated innumerable times eg bell type experiments.

Why is nature probabilistic? That's just the way she is. But as mentioned previously Gleason's theorem shows if you want to model observations by Hermitian operators that's something that's forced on us. Why Hermitian operators? Well as I mentioned above we want basis independence. But in explaining anything some assumptions must be made - these are the assumptions QM makes and it's been verified to an extremely high degree of accuracy.

Thanks
Bill