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## Wavefunction collapse: is that really an axiom

Can the wavefunction collapse not be derived or is it really an axiom?

How can the answer to this question (yes or no) be proven?

If it is an axiom, is it the best formulation, is it not a dangerous wording?

Let's enjoy this endless discussion !!!

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Blog Entries: 19 				 				 				 				         Recognitions:        Science  Advisor  There are several different interpretations of QM. In some of them, there is no need for a collapse postulate.

Recognitions:      Gold Member        Science  Advisor     Staff Emeritus  I think the thing we can sensibly say is that wavefunction collapse cannot follow from the unitary time evolution, which is easy to establish.

## Wavefunction collapse: is that really an axiom

Quote by vanesch 					 				 				I think the thing we can sensibly say is that wavefunction collapse cannot follow from the unitary time evolution, which is easy to establish.
But it is if you include the interaction with the measuring device into the QM model.

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Quote by lalbatros 					 				 				But it is if you include the interaction with the measuring device into the QM model.
Only if you choose some model other than unitary evolution for describing the measurement process.

Do you mean that the "measurement axiom" is contradictory to my/the postulate that the (unitary) equation of evolution governs all interactions? (including measurement sytems)

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Quote by lalbatros 					 				 				Do you mean that the "measurement axiom" is contradictory to my/the postulate that the (unitary) equation of evolution governs all interactions? (including measurement sytems)
yes, of course! That's the whole issue (or better, half the issue) in the "measurement problem". It is (to me at least) one of the reasons to consider seriously MWI.

There's no unitary evolution (no matter how complicated) that can result in a collapsed wavefunction. This can be shown in 5 lines of algebra.

Quote by vanesch 					 				 				yes, of course!  That's the whole issue (or better, half the issue) in the "measurement problem".  It is (to me at least) one of the reasons to consider seriously MWI.  There's no unitary evolution (no matter how complicated) that can result in a collapsed wavefunction.  This can be shown in 5 lines of algebra.
The algebra is simple and true.
The problem is that the wavefunction collapse doesn't really exist.
Just like microreversibility doesn't contradict the second law: microreversibility doesn't necessarily imply the existence of a chaos demon.

Quote by vanesch 					 				 				yes, of course!  That's the whole issue (or better, half the issue) in the "measurement problem".  It is (to me at least) one of the reasons to consider seriously MWI.  There's no unitary evolution (no matter how complicated) that can result in a collapsed wavefunction.  This can be shown in 5 lines of algebra.
The algebra is simple and true.
(even simple inspection of collapse algebra is enough for that, specially on the density matrix)
The problem is that the wavefunction collapse doesn't really exist.
Just like microreversibility doesn't contradict the second law: microreversibility doesn't necessarily imply the existence of a chaos demon.

In addition, I am quite sure that the collapse axiom can be derived from the Schrödinger equation. But the understanding is missing, to my knowledge.

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Quote by lalbatros 					 				 				In addition, I am quite sure that the collapse axiom can be derived from the Schrödinger equation. But the understanding is missing, to my knowledge.
There is actually a proof that it cannot. The proof is based on the fact that the Schrodinger equation involves only local interactions, while the collapse, including the cases with two or more entangled particles, requires nonlocal interactions.

There is, however, something that contains some elements of a collapse but can be obtained from the Schrodinger equation. This is the environment-induced decoherence. And it is closely related to the second law emerging from time-symmetric laws of a large number of degrees of freedom. See e.g.
http://xxx.lanl.gov/abs/quant-ph/0312059 (Rev. Mod. Phys. 76, 1267-1305 (2004))

Quote by vanesch 					 				 				This can be shown in 5 lines of algebra.
could you please show it then? i don't know about you guys, but i am so sick of qualitative arguments involving wavefunction collapse, etc.

to address the OP, it is my understanding is that if you postulate the Born interpretation then wavefunction collapse follows from that; in other words, physicists weren't just sitting around and postulated "wave collapse" as some popular books/shows would like one to believe.

more concretely,

$$\langle \Omega \rangle_\alpha = \sum_i \omega_i | \langle \omega_i | \alpha \rangle|^2$$

where the Born interpretation is that the quantity$$| \langle \omega_i | \alpha \rangle|^2$$ is to be interpreted as the probability amplitude of measuring a value $$\omega_i$$

$$\langle \Omega \rangle_\alpha = \sum_i \omega_i | \langle \omega_i | \alpha \rangle|^2$$
$$= \sum_i \omega_i \langle \omega_i | \alpha \rangle ^* \langle \omega_i | \alpha \rangle$$
$$= \sum_i \omega_i \langle \alpha | \omega_i \rangle \langle \omega_i | \alpha \rangle$$
$$= \sum_i \langle \alpha | \omega_i \rangle \omega_i \langle \omega_i | \alpha \rangle$$
$$= \sum_i \langle \alpha | \omega_i \rangle \langle \omega_i | \Omega | \omega_i \rangle \langle \omega_i | \alpha \rangle$$
$$= \langle \alpha |\left(\sum_i | \omega_i \rangle \langle \omega_i |\right) |\Omega |\left(\sum_i | \omega_i \rangle \langle \omega_i | \right)\alpha \rangle$$
$$=\langle \alpha | \Omega | \alpha \rangle$$

so for some state $$\phi = \sum_i c_i \psi_i$$ that is NOT an eigenket of the operator (but can always be formed from a linear combination of eigenkets), we have:

$$\langle \phi | \Omega | \phi \rangle = \langle \phi | \Omega | \sum_j c_j \psi_j \rangle$$
$$=\langle \phi | \sum_j c_j \omega_j \psi_j \rangle$$
$$=\sum_j \langle \sum_i c_i \psi_i | c_j \omega_j \psi_j \rangle$$
$$=\sum_i \sum_j c_i^* c_j \omega_j \langle \psi_i | \psi_j \rangle$$

and by orthogonality of states, we have:

$$=\sum_i |c_i|^2 \omega_i$$

which shows that the average value of our experiments will be a weighted average of the eigenstates, i.e. the "wavefunction collapse" s.t. any individual measurement will be a particular eigenvalue. in the classical limit, the spectra of eigenvalues is nearly continuous and so the effect is unnoticeable.

so whats the big deal?? can someone explain to me why this is, for some people, such a big damn mystery??

Quote by vanesch 					 				 				There's no unitary evolution (no matter how complicated) that can result in a collapsed wavefunction.  This can be shown in 5 lines of algebra.
There is no need for any algebra to show that the wavefunction collapse is not described by a unitary evolution. This follows simply from the definition of the wavefunction. By definition, the wavefunction is a probability amplitude. This means that the measurements described by the wavefunction is a random probabilistic unpredictable process, which cannot be described by deterministic "unitary evolution". That's the whole point of quantum mechanics. In my opinion, looking for a unitary description of the collapse is equivalent to looking for "hidden variables".

Eugene.

I agree, meopemuk. The collapse is not an unitary transformation, and it is even not a transformation at all.  After the collapse, there is no wave function anymore, but a statistical mixture.  That's the axiom.  My view is that after the interaction of a small system with a measuring device, the state of the small system loses its meaning,  and only the combined wavefunction has a meaning.  The problem that remains is how does the axiom emerge from the "complex" evolution. This is a challenge, and I am confident that it will or can be explained trivially. I am also sure that solving this problem is not really useful for the progress of QM,  that is it of the kind of problem that time and generations solves.

Recognitions:      Gold Member        Science  Advisor     Staff Emeritus  Ok, here goes the "proof".  Axiom 1: every state of a system is a ray in Hilbert space.  Now, consider the system "measurement device + SUT"  (SUT = system under test, say, an electron spin).  This is quantum-mechanically described by a ray in hilbert space.  As we have degrees of freedom belonging to the SUT and other degrees of freedom belonging to the measurement device, the hilbert space of the overall system is the tensor product of the hilbert spaces of the individual systems   H = H_m x H_sut  Now, consider that before the measurement, the SUT is in a certain state, say |a> + |b> and the measurement system is in a classically-looking state |M0>.  As we have now individually assigned states for each of the subsystems, the overall state is given by the tensor product of both substates:  |psi0> = |M0> x ( |a> + |b> )  Now we do a measurement.  That comes down to having an interaction hamiltonian between both subsystems, and from that interaction hamiltonian follows a unitary evolution operator over a certain time, say time T.  We write this operator as U(0,T), it evolves the entire system from time 0 to time T.  Now, let us first consider that our SUT was in state |a> and our measurement system was in (classically looking) state |M0>, which is its state before a measurement was done.  "doing a measurement" would result in our measurement device get into a classically looking state |Ma> for sure, assuming that |a> was an eigenvector of the measurement.  As such, our interaction between our system and our measurement apparatus, described by U(0,T) is given by:  U(0,T) { |M0> x |a> } = |Ma> x |a>  Indeed, the state of the measurement device is now for sure the classically-looking state Ma, and (property of a measurement on a system in an eigenstate) the SUT didn't change.  We can tell now the same story if the system was in state b:  U(0,T) { |M0> x |b> } = |Mb> x |b>  where Mb is the classically looking state of the measurement apparatus with the pointer on "b".  Now from linearity of U follows that:  U(0,T) { |M0> x (|a> + |b>) } = |Ma> x |a> + |Mb> x |b>  We didn't find "sometimes |Ma> x |a> and sometimes |Mb> x |b>" ; the unitary evolution gave us an entangled superposition.  Now, you can say "yes, but |u> + |v> means: sometimes |v> and sometimes |u>"  but that's not true of course.  Consider the other measurement apparatus N which does the following:  U(0,T) { |N0> x (|a> + |b>) } = |Nu> x (|a> + |b> )  U(0,T) { |N0> x (|a> - |b>) } = |Nd> x (|a> - |b>)  From this, we can deduce that   U(0,T) {|N0> x |a> } = 1/2 (|Nu> x { |a> + |b> } + |Nd> x { |a> - |b> })   Clearly if |u> + |v> means "sometimes u and sometimes v" then we could never have that |a> + |b> (which is then "sometimes a and sometimes b") always gives rise to Nu and never gives rise to Nd, because |a> gives rise to sometimes Nu and sometimes Nd.

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Quote by lalbatros 					 				 				I agree, meopemuk. The collapse is not an unitary transformation, and it is even not a transformation at all.  After the collapse, there is no wave function anymore, but a statistical mixture.  That's the axiom.
The problem is that the transition for a density matrix to go from "superposition" to "statistical mixture" is the following transformation:

take the densitymatrix of the "superposition", and write it in the matrix form in the *correct basis*. Now put all non-diagonal elements to 0. You now have the statistical mixture.

But again, that is a point-wise state change (if you take the density matrix to define the state) which is not described by the normal evolution equation of the density matrix. In other words, the transformation "superposition" -> "mixture" for the density matrix is again a state change which is not described by a physical interaction (which is normally described by the usual evolution equation of the density matrix).

In other words, that's nothing else but another way of writing down a non-unitary evolution, which is not the result of a known physical interaction.

My guess is that 99% of all experiments involve a single measurement of the system's state. (The counterexample is the bubble chamber, where we repeatedly measure particle's position and obtain a continuous track) In these cases we do not care what is the state of the systems and its wavefunction after the measurement (collapse). It is important to realize that one needs to consider the abrupt change of the wavefunction after measurement only in (not very common) experiments with repeated measurements performed on the same system.   Eugene.

Quote by vanesch 					 				 				The problem is that the transition for a density matrix to go from "superposition" to "statistical mixture" is the following transformation:  take the densitymatrix of the "superposition", and write it in the matrix form in the *correct basis*.  Now put all non-diagonal elements to 0.  You now have the statistical mixture.  But again, that is a point-wise state change (if you take the density matrix to define the state) which is not described by the normal evolution equation of the density matrix.  In other words, the transformation "superposition" -> "mixture" for the density matrix is again a state change which is not described by a physical interaction (which is normally described by the usual evolution equation of the density matrix).  In other words, that's nothing else but another way of writing down a non-unitary evolution, which is not the result of a known physical interaction.

I would rather say:

which is more conveniently described by a non-unitary transformation

Decoherence can already easily wipe off non-diagonal elements.

I also remember my master thesis 25+ years ago.
I worked on the Stark effect in beam-foil spectroscopy: the time-dependence with the quantum beats and the atomic decay.
(by the way the off-diagonal elements of the H-atoms exiting the foil were crucial in the simulation)

The hamiltonian had to simulate also the decays of the atomic levels.
Looks-like a non-unitary transformation too, isn't it?
I also didn't want to embarass myself with full QED stuff.
Guess how I modelled that:

- adding a non-hermitian term to the hamiltonian (related to the decay rates)
- and calculating the resulting non-unitary evolution operator
(the density matrix was therefore decaying, which is quite natural)

This is nothing strange or surprising and it shows clearly how non-unitary evolution can simply occur as a limit case of an unitary transformation.
With such a simple approach the Stark effect is very well calculated, for the energy levels, for the perturbed lifetimes and for the time-dependence and polarisations of light emission.

In somewhat pedantic words (I am not a mathematician): the limit of a series of unitary transformations may not be an unitary transformation, looks like that to me. And this can just mean that in some situations the unitary evolution may just be an academic vision: coarse graining makes it practical and non-unitary.

Why should I go for more science-fiction?

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