Mathematically what causes wavefunction collapse?

[bhobba]

There are two aspects which require some attention.
First, one must clarify the rationale for assigning a probability (which is a form of property) to a discrete occurrence of an event-type, better than assigning this probability to the event-type representing one category of events that may be observed when running the experiment.

I have zero idea what you are trying to say. Being able to assign probabilities to events is pretty basic and if it was in anyway not valid great swaths of applied mathematics from actuarial science to statistical mechanics would be in trouble - but they obviously arent.

I'm sorry I don't understand this last sentence, in particular what you say about the link between the occurrence of an event and the collapse of the wave function.

Its very simple:
http://en.wikipedia.org/wiki/Ensemble_interpretation

Thanks
Bill

 

[zonde]

Like most interpretations there are a number of variants. The one Einstein adhered to is the one presented by Ballentine in his book and the usual one people mean when they talk about it. And indeed it refers to an ensemble of systems exactly as I have been saying in this tread about the state referring to an ensemble of similarly prepared systems - its the one more or less implied if you want to look on probability the frequentest way.

If you say that measurement outcome is described by probability you say that the rule applies to individual event (relative frequencies emerge from statistical ensemble of idependent events). So you contradict what Einstein was saying.
You have to allow possibility that relative frequencies appear as certainty by deterministic physical process. And then it's Ensemble interpretation.

The assumption you make if you accept Gleason's theorem would go something like this - I don't know what outcome will occur but it seems reasonable I can associate some kind of probability to them.

I assume that assigning probability to outcome might lead to false predictions.

 

[audioloop]

Hi all, I was wondering mathematically ,what causes wave function collapse? and why does it exist in all it's Eigen states before measurement? Thanks for any help and please correct my question if I have anything wrong.

math is just description.


.

 

[craigi]

If you say that measurement outcome is described by probability you say that the rule applies to individual event (relative frequencies emerge from statistical ensemble of idependent events). So you contradict what Einstein was saying.
You have to allow possibility that relative frequencies appear as certainty by deterministic physical process. And then it's Ensemble interpretation.

Einstein wasn't saying that an ensemble is required, only that if we interpret QM as a desription of ensembles rather than individual events we avoid "unnatural" interpretations.

In my opinion, the term unnatural seems to have been used in order to make the statement correct, but also makes it completely subjective. For it to be objective he would've actually had to define what he means by unnatural and if I recall correctly this was effectively an expression of his frustration with indeterminism. He was asserting his own prejudices on nature. It would've been written from a faith position in local realist hidden variable theories. Which we now know to be invalid if we require counterfactual definiteness.

 

[Jilang]

You have to allow possibility that relative frequencies appear as certainty by deterministic physical process.

I disagree with this. If everything were determined by physics process how would you explain something like a decay rate for an atom or particle. These events have a probability but are inherently random or appear to be so.

 

[Sugdub]

I have zero idea what you are trying to say. Being able to assign probabilities to events is pretty basic and if it was in anyway not valid great swaths of applied mathematics from actuarial science to statistical mechanics would be in trouble - but they obviously arent.

I had a look to the Ensemble interpretation article you referred to and I must admit I found it anything but clear. The first section displays a quote by Einstein (reproduced in this thread in #57 by Zonde). I would be extremely surprised if in the original context Einstein used the word “system” in a different meaning than a “microscopic object”, I mean something less precise but in the same range as a “particle”. May be somebody could clarify this point.

In the second section of the same article, the “system” is defined as a single run of a quantum experiment, whereas an ensemble-system is defined as an iterative run of that experiment. That looks pretty similar to what I described in my previous inputs, although the use that is made of the word “system” makes the text quite harsh to digest. But then the key sentence according to which one should understand if and why the ensemble interpretation assumes that the wave-function is a property of one single iteration reads as follows:
“The ensemble interpretation may well be applied to a single system or particle, and predict what is the probability that that single system will have for a value of one of its properties, on repeated measurements”.
If “system” stands for “a single iteration of the experiment”, then the sentence actually assigns the “property” to the “repeated measurements” pattern, the ensemble-system, and not to a single run. If “systems” stands for a “microscopic system” (if not, the wording “system or particle” is irrational), then the sentence does not tell whether the property is assigned to a single run or not. The sentence does not include any justification anyway.
Further on an example is presented where a pair of dice, i.e. a physical object involved in the experimental device, plays the role of the so-called “system”. The ambiguity is maximal.

Let's make things simple. If one admits that the probabilistic property assigned to the iterative experiment reflects an underlying probabilistic property assigned to a more elementary level (the single iteration), then there is no reason why this second probabilistic property should not in turn reflect a third probabilistic property standing another level below, whatever the form it takes. This leads to a regression ad infinitum which can only stop when one specifies a level to which a deterministic property can be assigned. So the only realistic and credible alternative to stating that the property at the level of a single run is deterministic (which all physicists assume in the case of classical probabilities) is to accept that there is no property at all at this elementary level, so that the distribution pattern observed at the iterative level is a fundamental property which cannot be reduced to the appearance or synthesis of a more fundamental property.
I've explained in my previous input why and how the quantum formalism actually deals with transforming a distribution of relative frequencies into another distribution of the same nature, thanks to an appropriate mathematical representation using the orientation of a unit vector which makes the “amplitude of probability” an empty physical concept. The quantum formalism deals with a probabilistic property defined at the iterative level, reflecting the experimental truth.
Should there be a more fundamental property at a lower level, whichever level that means, then the quantum formalism would no longer be considered as the most fundamental theory dealing with quantum experiments. It would have to be replaced with a theory explicitly dealing with the lowest level property, and that property would necessarily be deterministic.

 

[bhobba]

If you say that measurement outcome is described by probability you say that the rule applies to individual event (relative frequencies emerge from statistical ensemble of idependent events). So you contradict what Einstein was saying.

That's simply not true.

It purely depends on your interpretation of probability. In the ensemble interpretation an observation selects an outcome from the conceptual ensemble and what that outcome is can only be described probabilistically.

In most versions of Copenhagen the state applies to an individual system, but is purely a representation of subjective knowledge about the outcome of observations.

Ballentine, correctly, in his book, points out, as Einstein did, the difficulty that arises if you consider it applies to something more definite that an ensemble (the collapse issue is the problem), but for some reason didn't consider the case where is was simply subjective knowledge, which is what most versions of Copenhagen think of the state as.

I assume that assigning probability to outcome might lead to false predictions.

But it doesn't.

Thanks
Bill

 

[bhobba]

Einstein wasn't saying that an ensemble is required, only that if we interpret QM as a desription of ensembles rather than individual events we avoid "unnatural" interpretations.

Exactly what Einstein was getting at is explained in Ballentine's book.

But basically its the collapse issue. The ensemble interpretation is one way out, considering it purely as a state of knowledge is another.

Also note, and it bears mentioning, Einstein did NOT disagree with QM as you will sometimes read - he considered it incomplete - not incorrect.

Thanks
Bill

 

[craigi]

I disagree with this. If everything were determined by physics process how would you explain something like a decay rate for an atom or particle. These events have a probability but are inherently random or appear to be so.

This would actually be pretty easy to construct a viable deterministic hidden variable theory for. Where they have problems, is when we consider separated entangled particles ans contexuality.

Classical systems that are considered fundamentally deterministic exhibit appararent randomness. In fact, a system that is fundamentally indeterministic can appear deterministic and vice versa.

Einstein believed that apparent indeterminism was fundamentally deterministic. I think that perhaps a better way to look at it, is how does determinism emerge so convincingly from indeterminism, in our experiences, that the human mind considers it to be so fundamental. There are indeterminstic processes taking place all around us on all scales, all the time, but we are much more atuned to the deterministic processes.

 

[bhobba]

I had a look to the Ensemble interpretation article you referred to and I must admit I found it anything but clear. The first section displays a quote by Einstein (reproduced in this thread in #57 by Zonde). I would be extremely surprised if in the original context Einstein used the word “system” in a different meaning than a “microscopic object”, I mean something less precise but in the same range as a “particle”. May be somebody could clarify this point.

In discussions about QM one often encounters an analysis of a typical measurement situation consisting of preparation, transformation, then measurement.

See figure 1 in the following for a discussion:
http://arxiv.org/pdf/quant-ph/0101012.pdf

Thanks
Bill

 

[zonde]

If you say that measurement outcome is described by probability you say that the rule applies to individual event (relative frequencies emerge from statistical ensemble of idependent events). So you contradict what Einstein was saying.

That's simply not true.

It purely depends on your interpretation of probability.

Interpretation does not change prediction, right? But if events are not independent we can get results that are quite different from predictions that are made using probabilities.

Do you agree?

As an example. Say we can have event + or - with equal probability (0.5). Now if we take series of events in a large sample we would expect that there will be series like ++++++++++ or ----------. And we can calculate how big a sample should be to expect series like that with say 99.99% probability.
But if events are not independent it is possible that series like ++++++++++ or ---------- can never appear (probability 0%) while relative frequencies for + and - is still 0.5 and 0.5.

 

[craigi]

But if events are not independent we can get results that are quite different from predictions that are made using probabilities.

Do you agree?

No.

Probability theory deals with correlated events perfectly well.

However, if you naively compute probabilities based upon an incorrect assumption of independence then your prediction will indeed be incorrect.

In fact, it's commonplace in physics to account for correlations to get the correct confidence interval for measurements.

See http://en.wikipedia.org/wiki/CovarianceIt's also worth noting that correlated probabilities in quantum mechanics and not just relevant to random errors in experiments, they're actually fundamental to the theory. If there were a problem with the prediction of quantum mechanics with respect to correlated events, someone would've definitely noticed by now!

 

[bhobba]

Interpretation does not change prediction, right?

Of course it doesn't.

But what it does do is change how you view it.

And indeed there is an assumption made in the Ensemble interpretation, and even the frequentest interpretation of probability, each trial is independent.

Its from the law of large numbers:
http://en.wikipedia.org/wiki/Law_of_large_numbers
'the expected value is the theoretical probability of success, and the average of n such variables (assuming they are independent and identically distributed (i.i.d.)) is precisely the relative frequency'

In modern times, as I have mentioned previously, the frequentest interpretation of probability is justified by the Kolmogorov axioms to remove any kind of circularity. As a byproduct it also justifies the Baysian view showing they are really different realizations of basically the same thing.

Thanks
Bill

 

[Jilang]

This would actually be pretty easy to construct a viable deterministic hidden variable theory for.

... Random particle decay. OK then, what sort of hidden variable (short of an inbuilt random number generator! ) do you think could achieve that?

 

[Superposed_Cat]

Random radioactive decay always has troubled me. I can handle the probabilistic nature of the wavefunction but decay has always bothered me.

 

[Maui]

Random radioactive decay always has troubled me. I can handle the probabilistic nature of the wavefunction but decay has always bothered me.

There are much more troublesome issues to be resolved especially wrt the foundations and spontaneous decay isn't one of them.

 

[stevendaryl]

... Random particle decay. OK then, what sort of hidden variable (short of an inbuilt random number generator! ) do you think could achieve that?

I'm not sure what's a plausible mechanism for particle decay, but there is no difficulty conceptually with assuming that it's deterministic. A sophisticated enough pseudo-random number generator, for example, is indistinguishable from a nondeterministic process.

What's difficult to accomplish with hidden variables is, as someone already pointed out, entanglement between distant subsystems.

 

[Jilang]

What's difficult to accomplish with hidden variables is, as someone already pointed out, entanglement between distant subsystems.

Well, it's certainly troubling me and the Cat!

 

[craigi]

... Random particle decay. OK then, what sort of hidden variable (short of an inbuilt random number generator! ) do you think could achieve that?

A pseudo random number generator.
http://en.wikipedia.org/wiki/Pseudorandom_number_generator

To be clear, I'm not arguing for a hidden varible theory, only that the decay of a particle is far from the greatest challenge for such a theory.

 

[Maui]

Random radioactive decay always has troubled me. I can handle the probabilistic nature of the wavefunction but decay has always bothered me.

They are not really 'particles' as you seem to imagine. The particle concept is a handy approximation. That's why spontaneous decay should be the last thing that bothers you. If this world were made of particles, atoms would have collapsed less than a second after the BB(less than a second after they were formed - some thousand years after the BB).

 

[PAllen]

Just thought I'd add here the clearest argument I've seen for "there is no problem in quantum mechanics". It will, of course, satisfy no one, but it is the clearest I've seen:

http://arxiv.org/abs/1308.5290

 

[Superposed_Cat]

I get that there is not really a problem per say with anything, I just have a minor problem with everything being based off probability. It used to be soothing to me last year but now it bothers me, and that decay is literally based off randomness(well exponential decay).

 

[craigi]

I get that there is not really a problem per say with anything, I just have a minor problem with everything being based off probability. It used to be soothing to me last year but now it bothers me, and that decay is literally based off randomness(well exponential decay).

I think once you get your head around the fact that determinism can emerge from indeterminism and vice versa, it doesn't seem that weird anymore. It happens in gases, weather systems and even economics, to name but a few.

At the moment, I'm not even sure that I see the concepts of determinism and indeterminsm as all that distinct anymore. Perhaps all we really have is a continuous scale with things that seem indeterministic at one end and things that seem deterministic at the other.

 

[Superposed_Cat]

I understand that, hence me previously being okay with it.
it's just that me and my friend were talking about the weirdness or things like the wavefunction, eulers theorem (we don't like complex numbers), t=0 of the big bang ect. It just bothers me that there are certain things we can't know as a result of physics.

Before discovering physics I accepted that you couldn't know everything in practice, but I don't like that we can never know certain thing regardless.

 

[craigi]

I understand that, hence me previously being okay with it.
it's just that me and my friend were talking about the weirdness or things like the wavefunction, eulers theorem (we don't like complex numbers), t=0 of the big bang ect. It just bothers me that there are certain things we can't know as a result of physics.

Before discovering physics I accepted that you couldn't know everything in practice, but I don't like that we can never know certain thing regardless.

Sometimes a question seems rational and but may in fact, be a meaningless question. That is not to say that it's wrong to ask it, only that question happens to have an illogical inconsistency already within it, that may not be immediately apparent.

The simplest example that I can think of to illustrate this is the question:

"what's north of the North Pole?"

Initially you may think that "nothing" is the correct answer, but when you think about it, the question is presuming there can exist more north than the maximum amount of north.

Another example might be:

"A man is standing somewhere in a room. What's in his lap?"
[If you're not a native english speaker, then "lap" may not translate too well.]

Again, if you're to answer "nothing", you're complicit in validating the question. The correct response is "a standing man doesn't have a lap".

In neither of these cases is nature conspiring to prevent us from knowing something. There is nothing to know. It is simply that we're asking a meaningless question. The same is true in physics. Often we are so bound by our experiences of the everyday world that we struggle to accept that the concepts that we use in it are not universally applicable.

 

[bhobba]

Just thought I'd add here the clearest argument I've seen for "there is no problem in quantum mechanics". It will, of course, satisfy no one, but it is the clearest I've seen:

http://arxiv.org/abs/1308.5290

Hmmm.

Interesting paper.

Have to say I agree with the following:
'Fifth, since neither decoherence nor any other mechanism select one particular outcome the whole “measurement problem” reduces to the question Why is there one specific outcome? which is asking Why are there randomly realized events? in the particular context considered. This harkens back to Sec. 1, where we noted that quantum theory cannot give an answer. In summary, then, the alleged “measurement problem” does not exist as a problem of quantum theory. Those who want to pursue the question Why are there events? must seek the answer elsewhere.'

Schlosshauer correctly identifies that as the key issue. Decoherence seems likely to answer all the other issues with the measurement problem - but that one it leaves untouched.

Is that a problem? Personally I don't know - I don't find it a worry - but I know others do.

What I do know is we have interpretations like DBB where it is not an issue at all and MWI where it has been replaced by something else. For me this suggests we have future surprises in store.

The following might be the beginnings of those surprises:
https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/

Only time will tell.

Thanks
Bill

 

[PAllen]

The following might be the beginnings of those surprises:
https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/

Only time will tell.

Thanks
Bill

I have been interested in that from popular presentations like you link. Unfortunately (for me) there is a bunch I need to learn to try to understand this work in a meaningful way.

 

[bhobba]

I have been interested in that from popular presentations like you link. Unfortunately (for me) there is a bunch I need to learn to try to understand this work in a meaningful way.

Indeed.

But, if what it reports is true, that they are replacing unitary evolution with something else it could have big consequences for the measurement problem - but of course only time will tell.

Thanks
Bill

 

[zonde]

Of course it doesn't.

But what it does do is change how you view it.

But it does not change the assumption that each trial is independent, right?

And indeed there is an assumption made in the Ensemble interpretation, and even the frequentest interpretation of probability, each trial is independent.

That contradicts that Einstein quote about ensemble interpretation and QM being not applicable to individual systems (trials).

 

[bhobba]

But it does not change the assumption that each trial is independent, right?

Its the assumption of the law of large numbers.

That contradicts that Einstein quote about ensemble interpretation and QM being not applicable to individual systems (trials).

I have zero idea why you say that. Its simply not true.

The logic is dead simple. By the law of large numbers we can find an ensemble associated with an observation where the proportion of outcomes is the probability. This follows from simply assuming the outcome can be described probabilistically. The state is not even introduced at this point. The Ensemble Interpretation associates the state not with individual systems but with the ensemble. Its that easy. If you still don't get it I will have to leave it to someone else because I simply can't explain it any better.

Thanks
Bill