Measurements and wavefunction collapse.

[maka89]

Hey everyone.I have a couple of question regarding wave function collapse.

I can accept that one cannot make a measurement with absolute precision and have a usable wave function afterwards due to the uncertainty principle...

Consider a particle moving along the x-axis with a wave function that has the probability-distribution of a moving gaussian curve. After some time, the wave function has become so smeared out that and has a so big std. dev. that it is not practical to work with it anymore.

1: Is there some operator or some maneuver that when used on the wave function, reshapes it into a wave function with a smaller standard deviance, but with a different mean(chosen probabilisticly) than the original? (Measuring the the particle, but accepting/forcing some uncertainty in the measurement)

2: If not: How do you think of an actual real physical measurement of a particle?

 

[rigetFrog]

Yes. I believe if you work out the mathematics for the double (or single) slit, the wave function will be reduced in size to the vicinity of the slit (or slits).

From a general hand waving mathematical perspective, the potential well associated with narrow slit is broadly distributed in reciprocal (k) space. When this potential interacts with the e-, it imparts momentum that broadens the e- wave function in reciprocal space, which corresponds to a narrowing in real space.

If I find a good source, I'll post it later.

 

[bhobba]

I can accept that one cannot make a measurement with absolute precision and have a usable wave function afterwards due to the uncertainty principle..

Well actually that isn't quite what QM says - it places no limit on the precision of any measurement - what it places a limit on is the precision of certain measurements that can be made at the same time. In practice of course, even though in principle there is no reason why you can't make a measurement with 100% accuracy, that's not possible. That observation is important regarding certain technical aspects of the theory concerned with things like the Dirac delta function and Rigged Hilbert Spaces and making sense of it but that's a story for another time.

Is there some operator or some maneuver that when used on the wave function, reshapes it into a wave function with a smaller standard deviance, but with a different mean(chosen probabilisticly) than the original? (Measuring the the particle, but accepting/forcing some uncertainty in the measurement)

Sure - you simply use a measuring device with the precision you want the new variance to be. An example would be the slit in the double slit experiment - you vary the precision of the location of the object going through the slit by adjusting the width of the slit.

If not: How do you think of an actual real physical measurement of a particle?

The way to think of an actual measurement, any measurement, is to go back to what an observation is. It will have some outcomes yi. The fundamental axiom of QM is those yi are associated with a set of disjoint positive operators Ei ∑ Ei = 1 called a resolution of the identity. You can combine those into an operator O = ∑yi Ei which is a Hermitian operator and via the spectral theorem you can recover the yi and Ei. O is called the observable associated with the operator.

Of course if we do the observation many times we will get an expected value E(O). That is determined by the so called Born rule, but just for the heck of it I will derive the Born Rule from a simple assumption to show its not something just pulled out of a hat. The assumption is its linear ie if O1 and O2 are observables E(c1*O1 + c2*O2) = c1*E(O1) + c2*E(O2).

First its easy to check <bi|O|bj> = Trace (O |bj><bi|).

O = ∑ <bi|O|bj> |bi><bj| = ∑ Trace (O |bj><bi|) |bi><bj|

Now we use our linearity assumption

E(O) = ∑ Trace (O |bj><bi|) E(|bi><bj|) = Trace (O ∑ E(|bi><bj|)|bj><bi|)

Define P as ∑ E(|bi><bj|)|bj><bi| and we have E(O) = Trace (O P).

P, by definition, is called the state of the quantum system. The following are easily seen E(1) = 1 so Trace (P) = 1. Thus P has unit trace. E(|u><u|) from the definition of an observable (since the outcomes are 0 and 1) is a potitive number >= 0. Thus Trace (|u><u| P) = <u|P|u> >= 0 so P is positive.

So we have the Born rule which says a positive operator of unit trace P exists such that the expected value of an observation O is Trace (PO). P is called the state of the system.

The point of the above is to bring home that the state of the system is not necessarily (it may be - but it doesn't have to be) something physical like an electric field - it, like probabilities, is simply something that helps us to calculate the expected value of observation.

Just as an aside Von Newuman used a similar argument to show hidden variables did not exist - the error he made however is hidden variables do not have to obey the linearity assumption. A deeper analysis also shows that linearity depends crucially on non contextuality as shown by an important theorem called Gleason's theorem - but that is a story for another time.

Thanks
Bill

 

[bhobba]

If I find a good source, I'll post it later.

In that connection the following may be of interest:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

Thanks
Bill

 

[strangerep]

How do you think of an actual real physical measurement of a particle?

First, one must understand the difference between a filtering operation, (in which an initial state becomes a new state), and a measurement where the initial state is essentially obliterated (such as detection of a particle hitting a screen).

Ballentine ch9 explains a more general framework for realistic measurement -- in terms of interaction between an initial object state and an apparatus. In general, the post-interaction of the apparatus is correlated with the object's pre-measurement state.

 

[bhobba]

Ballentine ch9 explains a more general framework for realistic measurement -- in terms of interaction between an initial object state and an apparatus. In general, the post-interaction of the apparatus is correlated with the object's pre-measurement state.

An excellent source, not only for that but seeing how QM is simply the working out of what an observable is and the Born rule. Highly recommended.

Apart from that book the following will also help:
http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

It discusses how POVM's come about, which are a generalisation of resolutions of the identity - it basically has the disjoint requirement removed ie a quantum measurement is described by a set of positive operators Ei that sum to one. Its from a system being measured interacting with a probe and the probe observed.

Interestingly the proof of the Born rule for POVM's is quite easy without an assumption of linearity, but simply assuming non contextuality ie all you need to assume is the probability of outcome i depends only on Ei and not what POVM it is part of.

Puts what a quantum state is in a new and interesting light.

Thanks
Bill

 

[atyy]

1: Is there some operator or some maneuver that when used on the wave function, reshapes it into a wave function with a smaller standard deviance, but with a different mean(chosen probabilisticly) than the original? (Measuring the the particle, but accepting/forcing some uncertainty in the measurement)

Maybe http://arxiv.org/abs/math-ph/0512069 or http://arxiv.org/abs/quant-ph/0611067 ?