When does the wavefunction collapse?

[Sam_Goldberg]

Hi guys,

Quantum mechanics gives well-defined probabilistic predictions for the value we get when we measure position or momentum; one simply takes the absolute square of the wavefunction in either the x-basis or the p-basis. However, I am not so clear on how we would predict at what time we obtain the result, and whether the answer be definite or probabilistic, we should be able to answer this question. I have found and read this (relatively short) article: http://arxiv.org/abs/quant-ph/9802020 and would much appreciate it if you guys could read it and help answer a few questions I have.

In the article, Rovelli defines the M operator that acts on the S-O system. He says that the question: when does O see a definite value of a dynamical variable? can be answered probabilistically by using a second observer O' and measuring the value of the observable corresponding to the M operator. If it's 1, the measurement has been made, and if it's 0, the measurement has not been made. This is a probabilistic answer to the question, because it involves measuring a quantum mechanical operator.

I cannot comprehend this for the following reason: it seems as if we can define the M operator in many different ways. By defining M differently, we predict that with some other probability, the wavefunction of S collapses for the observer O under a different kind of projection operator. To put it really concisely: by suitably redefining the M operator, we predict a different type of wavefunction collapse, with a different probability in time.

Furthermore, in real life, S and O are always weakly interacting, and I believe that the singular value decomposition theorem says that we can always find a set of basis vectors for the S-O system such that S and O are perfectly correlated. Even though the correlation might be for some operators corresponding to totally uninteresting observables, the correlation is still there. Then if we, again, suitably define the M operator, would we not predict (with certainty) that O sees S undergo a wavefunction collapse? Then would we not have wavefunction collapse all the time? True, the interaction between S and O is weak, and so the collapse means that the wavefunction of S with respect to O barely changes (I hope). However, one still wonders whether this would affect normal Schrodinger evolution. A really stupid sounding but serious question: if collapse happens all the time (continuously), then is there any "opportunity" for Schrodinger evolution to take place?

As you guys can see, I'm pretty confused, so any help would be much appreciated. I'm really trying to understand this article. Thanks.

 

[Sam_Goldberg]

Hm... no replies. Perhaps if I suitably redefine the M operator, I can get the wavefunction to collapse and people will answer my question. Oh wait! The M operator is defined in the context of the O' system, and my viewpoint is from the O system, so actually, I can't do a thing about it.

In all seriousness though, if nobody wants to read the short article I provided a link to, then I would at least like to know if anyone knows the answer to the basic question I have been asking: how can we probabilistically determine when the wavefunction collapses? If we can probabilistically determine the result of a position or momentum measurement, we should be able to figure out when we will obtain the result. If not, that seems like a serious deficiency of quantum mechanics, right?

 

[strangerep]

Sam,

Possibly the reason you didn't get any answers initially is that the
notion of wave function collapse is part of an old QM interpretation
which is becoming increasingly deprecated in modern times.

Try studying Ballentine's book "QM -- A Modern Development", in which
he discusses the "statistical interpretation", and also gives some
examples of where the old collapse interpretation has problems.

 

[Sam_Goldberg]

Thanks for the book suggestion. Unfortunately I'm on break before my school starts, so I don't have access to a library with tons of good physics books. I will look at the book when I get back, however.

Nevertheless, if it is possible, I'm interested in hearing what the statistical interpretation you mentioned says about this example. Let's fire an electron through a single slit. The wavefunction diffracts, and we see a point light up on a film behind the single slit. This is, to my knowledge, what would happen in real life. How would we be able to predict when we see the flash, according to the statistical interpretation?

I know you mentioned that the idea of the wavefunction collapse is becoming less trusted. Nevertheless, if in "real life" we see the film light up at one point in spacetime, how would we predict the value of t? For the value of x, it is clear, of course; all the textbooks on quantum mechanics have an answer to that question. But for the value of t, I'm not sure.

 

[arkajad]

Perhaps you will not be particularly impressed, nevertheless a possible mechanism of determining the time of the collapse is described in this paper:

http://arxiv.org/abs/quant-ph/9602010"

If you will have any questions concerning this particular approach - I can answer them.

 

[calhoun137]

There is an answer to the question in your title, and it is this:

The wave function will collapse whenever a quantum system interacts with a system which obeys classical mechanics to a sufficient approximation.

This is all spelled out very clearly in Landau vol 3, chapter 1.

It might simply be the case that you have discovered an error in this paper; I haven't looked very carefully at it. I think the idea of a wave function collapsing is so philosophical that the idea of measuring it's timing is questionable; so I agree with the previous poster who pointed out how poorly defined this notion is. For every crazy idea about the philosophy of quantum mechanics, you could probably write a paper on how it explains the timing of the wave function collapse. I want to see is an experiment that can measure this timing; and personally, I find a purely theoretical paper on this subject hard to take seriously.

My interpretation is that the timing of the wave-function collapse, if it existed, would be a hidden variable i.e. a physical quantity which is not measurable in principle due to the the nature of quantum mechanics. There is a theorem which states there can be no hidden variables; so therefore, in my view, the concept of the "timing" of the wave function collapse does not have a physical meaning. Another way to see this is, is that in quantum field theory the temporal order of events over very short periods of time does not have a physical meaning (see Landau vol 4, but what I am saying is basically a statement of the Feynman interpretation of quantum mechanics, i.e. the probability amplitude is the sum over the amplitude for all possible ways that event can happen, regardless of temporal order)

 

[arkajad]

I want to see is an experiment that can measure this timing...

I think this is a very good idea!