Mathematically what causes wavefunction collapse?

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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.

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Nothing in the Mathematical formalism of QM does it predict wave function collapse.

But where does the idea come from? and what of the double-slit experiment?

kith
Hi all, I was wondering mathematically, what causes wave function collapse?
It's just a heuristic rule. If you perform a measurement, you find the system in an eigenstate to the corresponding observable and the probability for this is given by the Born rule. So the Copenhagen interpretation introduces the collapse rule which does exactly this.

Many people dislike collapse because of this. There are numerous interpretations of QM which don't need collapse but all of them are weird some other way.
and why does it exist in all it's Eigen states before measurement?
This depends on the observable. Your state is an eigenstate wrt to some observables and a superposition wrt to other observables. Such observables are called incompatible with the first set of observables. Their existence is the cause of Heisenberg's uncertainty principle.

But where does the idea come from? and what of the double-slit experiment?
The idea comes from that we see a definite result (i.e spin up or down), not a superposition of spin up and spin down.

I suggest you read up on the measurement problem. Wikipedia isn't too bad at explaining that.

There are also some good chapters in David Albert's "Quantum Mechanics & Experience" on the issue.

I get the heuristics an the intuition (what little there is) I was just hoping for something more concrete mathematically.

Hall (quantum theory for mathematicians) treats wave function collapse as an axiom of quantum mechanics.

"Suppose a quantum system is initially in a state ψ and that a
measurement of an observable f is performed. If the result of the measurement
is the number λ ∈ R, then immediately after the measurement, the
system will be in a state ψ' that satisfies fψ=λψ' "

kith
You could read a bit about decoherence. It helps to state the problem much clearer and it explains why some interpretations don't need collapse.

QM suggests an answer to the question why collapse is only a heuristic rule. If you perform a measurement, you get entangled with the system. The resulting state is a superposition of "you experiencing A" and "you experiencing B". The Copenhagen interpretation says that the real experience is selected by collapse. The Many Worlds interpretation says both experiences are real, they belong to different worlds. Therefore, it doesn't need collapse.

The introduction of collapse could be seen as sticking to reductionism while the QM math suggests a more holistic picture, where the experimenter and system can't be separated.

Avodyne
Understanding "collapse" requires mathematical modeling of the entire system, including the measuring apparatus. This is not easy to do, and drastic approximations are typically made. There is a very large literature on this. Here is just one (relatively user friendly) paper: http://arxiv.org/abs/quantph/0306072

Nugatory
Mentor
I get the heuristics an the intuition (what little there is) I was just hoping for something more concrete mathematically.

Seriously though, as far as the math of QM goes, it doesn't get much better than the postulate that every observable corresponds to a Hermitian operator, and that a measurement of that observable must yield a value that is an eigenvalue of the corresponding operator.

especially the derivations

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bhobba
Mentor
You need to study Ballentine - QM - A Modern Development:

Here you will find QM developed from just two axioms and why the outcome of an observation is an eigenvector, and indeed what a state is in the first place. Schrodinger's equation, for example, is given its correct basis - symmetry.

And once you have grasped that then you can take a look at Gleason's Theorem and see that the second axiom more or less follows from the first:
http://kof.physto.se/theses/helena-master.pdf

After that you will understand exactly what QM is about, and its true foundational issue encoded, basically, in just one axiom.

The reason its not usually presented this way is the math is FAR from trivial. But its really the only way to understand just what the theory says.

Then when you have finished that you can look into decoherence which is the basis of much of the modern interpretations of QM:
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

The measurement problem has not been solved - the collapse issue is still there in modern treatments, but decoherence has explained APPARENT collapse, which for many people, myself included, is good enough.

If you want to go even deeper into it get my go-to book on it by Schlosshauer
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

And after all that if you want our very deepest and most sophisticated version of QM then check out:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Be warned however - such books are called by mathematicians non trivial - which is a euphorism for HARD.

Thanks
Bill

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bhobba
Mentor
Seriously though, as far as the math of QM goes, it doesn't get much better than the postulate that every observable corresponds to a Hermitian operator, and that a measurement of that observable must yield a value that is an eigenvalue of the corresponding operator.
That's the first axiom in Ballentine's treatment, the second axiom following from that via Gleason's theorem, so basically that's it, that's all - just one axiom.

This is the fundamental foundational postulate from which all of QM basically follows - and Ballentine gives its detail.

But what it means - that is a MUCH MUCH more difficult matter.

Still its very wise to understand the mathematical formalism and exactly what its axiomatic basis is before delving into that minefield.

And it will take you WAY beyond, well to be blunt, the sickening tripe often found in the popular press that use QM to promote mystical nonsense like What the Bleep Do We Know Anyway.

And finally here is the way I like to look at that single axiom.

Imagine we have a system and some observational appartus that has n possible outcomes associated with values yi. 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. This is the foundational axiom of QM, and heuristically why its resonable.

If you want an even deeper foundational treatment based on the modern view, nowadays its often thought of as just a novel version of probability theory - there basically being just two reasonable models applicable to physical systems. Check out:
http://arxiv.org/abs/quant-ph/0101012
http://arxiv.org/abs/0911.0695

That would probably be the most recent view - QM is basically a probability model - there are many of those and the study of such is a modern development - but for modelling physical systems some very reasonable assumptions leads to basically two - bog standard probability theory you learnt about at school and QM - but what distinguishes QM is it allows entanglement, which would seem the rock bottom, basic, essential wierdness of QM.

Thanks
Bill

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There are many sources but you are not ready for those. You should likely begin with a more elementary treatment (after you finish calculus and linear algebra).

So I can say I did answer your question though, Ballentine is the standard mathematically oriented quantum book. A more elementary but still slightly more mature book than griffiths is Zettili, quantum mechanics.

meBigGuy
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vanhees71
Gold Member
2019 Award
In the mathematical formalism of QM is no collapse of the state, and it's not necessary at all to claim that there is one. It's only one flavor of the Copenhagen interpretation of quantum mechanics, and it causes a lot of trouble, particularly inconsistencies with causality in relativistic quantum theory.

For a good exposition of the foundations of quantum theory, I'd also recommend to read Ballentine's textbook, which follows the Minimal Statistical Interpretation, which has been already mentioned in this thread, or also the newest textbook by Weinberg, who gives a good overview over some of the interpretations of quantum theory too:

S. Weinberg, Lectures on Quantum Mechanics, Cambridge University

bhobba
Mentor
In the mathematical formalism of QM is no collapse of the state, and it's not necessary at all to claim that there is one. It's only one flavor of the Copenhagen interpretation of quantum mechanics, and it causes a lot of trouble, particularly inconsistencies with causality in relativistic quantum theory.
Indeed.

Few people seem to mention that - don't quite know why.

That's just another reason to study Ballentine - he explains it all carefully - its one of the few books that does.

Heard good things about Wienberg's book as well - but don't personally have it - although it's on my list.

Thanks
Bill

It might be too obvious but it's worth repeating - the mathematical formalism does not explain what we classical observers see(single outcomes). It's a crippled model and needs additional fancy stuff, hence the need for collapse postulates, unobserved universes and magical guiding waves. In other words, even if you choose to look the other way, the measurement problem is still there.

Many people dislike collapse...... There are numerous interpretations of QM which don't need collapse but all of them are weird some other way.
All of the interpretations are attempts to explain a fundamental unknown: why measurements are statistical at the quantum level. ALL are 'weird' because quantum mechanics is.

http://plato.stanford.edu/entries/qt-measurement/

From the inception of Quantum Mechanics (QM) the concept of measurement proved a source of difficulties that found concrete expression in the Einstein-Bohr debates, out of which both the Einstein Podolsky Rosen paradox and Schrödinger's cat paradox developed. In brief, the difficulties stemmed from an apparent conflict between several principles of the quantum theory of measurement. In particular, the linear dynamics of quantum mechanics seemed to conflict with the postulate that during measurement a non-linear collapse of the wave packet occurred.

The dynamics and the postulate of collapse are flatly in contradiction with one another ... the postulate of collapse seems to be right about what happens when we make measurements, and the dynamics seems to be bizarrely wrong about what happens when we make measurements, and yet the dynamics seems to be right about what happens whenever we aren't making measurements. (Albert 1992, 79)
Wikipedia says this:

The Schrödinger equation provides a way to calculate the possible wave functions of a system and how they dynamically change in time. However, the Schrödinger equation does not directly say what, exactly, the wave function IS. Interpretations of quantum mechanics address questions such as what the relation is between the wave function, the underlying reality, and the results of experimental measurements.
QM suggests nature is fundamentally indeterministic, meaning nature exhibits statistically based observables. A quantum system is described by a quantum state. The evolution in time of a state is described by the wave function: but there is no universal on what the wavefuntion means let alone it's possible 'collapse'. The effect of a measurement on the state [wave function] makes it jump into some eigenstate…but which eigenstate is a matter of chance!!

And I especially like this:
The following quote is from Roger Penrose celebrating Stephen Hawking’s 60th birthday in 1993 at Cambridge England.....this description offered me a new insight into quantum/classical relationships:

Either we do physics on a large scale, in which case we use classical level physics; the equations of Newton, Maxwell or Einstein and these equations are deterministic, time symmetric and local. Or we may do quantum theory, if we are looking at small things; then we tend to use a different framework where time evolution is described.... by what is called unitary evolution...which in one of the most familiar descriptions is the evolution according to the Schrodinger equation: deterministic, time symmetric and local. These are exactly the same words I used to describe classical physics.

However this is not the entire story..... In addition we require what is called the "reduction of the state vector" or "collapse" of the wave function to describe the procedure that is adopted when an effect is magnified from the quantum to the classical level.....quantum state reduction is non deterministic, time-asymmetric and non local....The way we do quantum mechanics is to adopt a strange procedure which always seems to work...the superposition of alternative probabilities involving w, z, complex numbers....an essential ingredient of the Schrodinger equation. When you magnify to the classical level you take the squared modulii (of w, z) and these do give you the alternative probabilities of the two alternatives to happen...it is a completely different process from the quantum (realm) where the complex numbers w and z remain as constants "just sitting there"....in fact the key to keeping them sitting there is quantum linearity...

bhobba
Mentor
the measurement problem is still there.
It's there in one form or another, meaning each interpretation handles it its own way, and not everyone agrees which is the best way.

IMHO that's the real issue with QM - that each interpretation sucks in its own unique way - not any particular issue such as what causes wave-function collapse because for a particular issue one interpretation has there is another where it doesn't even exist or is a non issue.

Thanks
Bill

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atyy
Hi all, I was wondering mathematically ,what causes wave function collapse?
Collapse is what happens when you make certain sorts of measurements, and is simply a postulate. The unsatisfying thing about this postulate is that there seems to be two sorts of time evolution. When no measurement is made, the state evolves unitarily according to Schroedinger's equation. When a measurement is made, the state does not evolve unitarily, but instead it collapses. Also, a textbook answer as to what a measuring apparatus is, is that it is a classical apparatus. If quantum mechanics is more fundamental than classical physics, it seems unsatisfying that classical objects are needed in the postulates of quantum mechanics. Nonetheless, collapse works and is consistent with experiments.

The collapse postulate is #3 in http://www.theory.caltech.edu/people/preskill/ph229/notes/chap2.pdf and #4 in section III of http://arxiv.org/abs/0903.5082.

Some people prefer a different measurement postulate. You can find it in http://en.wikipedia.org/wiki/POVM, Nielsen and Chuang's Quantum Computation text or section 3 and 6 of http://arxiv.org/abs/1308.5290, the latter of which says "The folklore that “a measurement leaves the system in the relevant eigenstate of the observable” applies only to over-idealized projective measurements (meaning that the Kks are pairwise orthogonal projectors). It is puzzling that some textbook authors consider it good pedagogy to elevate this folklore to an “axiom” of quantum theory.". But there is still "collapse" or "state reduction" here.

If you find collapse unsatisfying, you can explore http://arxiv.org/abs/quant-ph/0312059.

and why does it exist in all it's Eigen states before measurement?
A state is a vector. An observable corresponds to an operator. The eigenvectors of the operator form a basis, ie. an arbitrary vector can be written as a weighted sum of the eigenvectors. So an arbitrary state can be written as a sum (ie. superposition) of eigenvectors.

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bhobba
Mentor
Also, a textbook answer as to what a measuring apparatus is, is that it is a classical apparatus.
Actually quite a few textbooks don't make that clear at all - they should - but don't - in fact they leave it it up in the air what an observation is and you have people thinking its when a conscious observer registers it. IMHO this is the cause of a great deal of confusion.

Good texts like Ballentine make it clear exactly what an observation is - but that is not always the case.

Thanks
Bill

The confusion is bound to continue as there exist no classical apparatuses. They are all quantum in nature as is everything else in this reality.

bhobba
Mentor
The confusion is bound to continue as there exist no classical apparatuses. They are all quantum in nature as is everything else in this reality.
Its is an assumption that such exist.

How it emerges is a question that's under active research, but a lot of progress has been made.

But that's not the point - the point is by not explicitly stating the assumption it leads to a lot of confusion. The best books like Ballentine make it very clear - but not all are that careful.

Thanks
Bill

vanhees71
Gold Member
2019 Award
I still don't understand, why some people claim you need a collapse of states to interpret quantum theory and why you need an explanation of the probabilistic nature of quantum-theoretical predictions. This is just one of the postulate known as Born's rule. The point is that this gives a consistent picture and an overwhelming empirical success for a very broad range of observations (in fact there is no evidence whatsoever that quantum theory is violated by any empirical fact). That's what a good physical theory should do, and you can't expect more. The question, why Born's rule holds true and why the description of nature on a fundamental level is indeterministic is not asked in the realm of physics. You may wonder about it and try to find a simpler or more intuitive set of postulates defining quantum theory (e.g., Weinberg discusses at length, whether Born's postulate can be derived from the other postulates, i.e., the usual kinematical and dynamical postulates in terms of the Hilbert-space formulation with observable operators and state operators, coming to the conclusion that it cannot be derived), but as long as there is no empirical evidence against quantum theory, you better keep this theory.

The minimal interpretation just takes the mathematical formalism and gives the minimal interpretation to enable us to use it for predictions of the real world. The operational definition of a state is (an equivalence class of) a prepartion procedure, leading to the description of the system's state with a given (pure or mixed) state in the quantum-mechanical formalism. This together with an observable algebra, particularly including a Hamiltonian of the system, encodes everything you can know about the system within the rules of quantum theory. Among other things you know, which observables have a determined value and which don't and with which probability you find some value of any observable when it's measured. Sometimes you can even say which new state describes the system after a measurement. Then you can take the measurement (and appropriate filtering if necessary) as a possible preparation procedure for this state.

Another very nice book about all this, I forgot to mention before, is

A. Peres, Quantum Theory: Concepts and Methods, Kluwer Academic Publishers