Why doesn't wave function collapse contradict Schrodinger equation?

[andrewkirk]

The postulates of quantum mechanics include:

(1) Schrödinger's equation describes how the wave function of a system changes over time, and appears to make the wave function continuous over time.

(2) When a measurement is made of quantity m, the wave function instantly changes to an eigenvector of the corresponding operator M. It seems that such a change would most likely be discontinuous in time.

It seems as though 2 contradicts 1, unless 1 were stated differently, eg 'Schrödinger's equation describes how a wave function changes over time, between observations'. But the presentations of the postulates I have read don't state it that way, unless I missed it.

How can Schrödinger's equation be reconciled with wave function collapse?

Thanks for any answers!

 

[homeomorphic]

There is a contradiction--sort of. One of the mysteries of QM. But it works.

The reconciliation is just that Schrödinger's equation isn't supposed to be used when you make a measurement. When you measure, you use the projection postulate or some such thing.

So, really, the contradiction is just part of the theory. You have one rule (Schrödinger's equation) that describes time evolution, and another rule than describes measurement.

In some sense, it's not contradictory. It's just that the theory requires different rules for different situations. Some situations require Schrödinger evolution and some require state reduction. It's weird for a fundamental physical theory to require that, but it works. The big question is what, if anything, is really going on behind the scenes there. No one knows.

 

[tom.stoer]

There are also no-collaps interpretations trying to avoid this artifiial collaps and the strange separation between a quantum system and a classical experiment. I would say that today the orthodox Copenhagen interpretation is no longer mainstream.

 

[cmos]

In my opinion, there is no contradiction.

For this discussion, you should also include the postulate that states that a system is completely described by its wavefunction. By your first postulate, Schrödinger's equation then gives the wavefunction's evolution up to the point that you make your measurement. Why do we then have "wavefunction collapse?"

One way of interpreting it is that you have a collapse of the wavefunction because you have introduced your measurement apparatus. Introducing this new component into the original system changes the system and thus the original wavefunction is no longer valid; it is no longer valid for at least for as long as your measurement apparatus takes to obtain the measurement.

So your statement

unless 1 were stated differently, eg 'Schrödinger's equation describes how a wave function changes over time, between observations'. But the presentations of the postulates I have read don't state it that way, unless I missed it.

seems to be the best way of thinking about your problem with the caveat, however, that the Schrödinger equation still holds even during your observation -- you just need to use the new wavefunction that includes the interaction of your measurement apparatus.

So I would say that the part that you missed was that you have actually have two systems: one without and one with the measurement apparatus.

 

[tom.stoer]

In my opinion, there is no contradiction.

At least in the orthodox Copenhagen interpretation there is an 'intrinsic contradiction' (refer to homeomorphic's post) b/c the time-evolution by the Schrödinger equation is unitary, whereas the collaps is non-unitary and therefore cannot be described in terms of quantum mechanics. So QM according to the orthodox Copenhagen is either self-contradictory or incomplete.

 

[dextercioby]

One of the postulates of classical mechanics is the principle of separation of interactions (forces). Thus the interaction between the Earth and the Sun is by no means influenced/changed by the interaction between sun and Venus or Venus and Earth. Thus one is free to leave aside the other planets and consider only the motion of the Earth & Sun as if they were the only celestial bodies in the universe.

Apparently one would like to extend this principle to quantum mechanics as well. Namely the system(s) are considered in interaction one with another, these interactions can be separated just like in classical mechanics. This is ok so far. But then comes the issue of measurement. Namely one forces into the theory of quantum systems the presence and the influence of the measurement apparatus, thing which is completely left out of classical mechanics. In its initial formulation, the so-called projection postulate proposed by John von Neumann was accepted by everyone in the community, even though, it contradicted the proposed axiom regarding the evolution of quantum states through the Schrödinger equation.

To avoid this contradiction, one can completely dismiss the projection postulate (see the work of Ballentine), or take the Hamiltonian in the Schrödinger equation to include the interaction between the apparatus of measurement and the system's observables being measured. But then the evolution of the quantum state with this bigger Hamiltonian would be determined and no projection postulate would be necessary, since it would be a mere theorem from the new Schrödinger equation, or...??

Not to mention, is the collapse unitary ??

 

[Fredrik]

As far as I can tell, there is no contradiction if you take the time evolution postulate to say that the wavefunction of an isolated system is described by the Schrödinger equation. Since no system is completely isolated from its environment, perhaps we should say something like this instead: The wavefunction of a system with negligible interactions with its environment is approximately described by the Schrödinger equation as long as the interactions with the environment remains negligible.

The wavefunction can change with time in a different way during measurements, because they are by definition non-negligible interactions between the system and a part of its environment.

 

[tom.stoer]

As far as I can tell, there is no contradiction if you take the time evolution postulate to say that the wavefunction of an isolated system is described by the Schrödinger equation. Since no system is completely isolated from its environment, perhaps we should say something like this instead: The wavefunction of a system with negligible interactions with its environment is approximately described by the Schrödinger equation as long as the interactions with the environment remains negligible.

The wavefunction can change with time in a different way during measurements, because they are by definition non-negligible interactions between the system and a part of its environment.

That's not consistent.

Decoherence shows that even the interaction with the environment can be treated according to QM, therefore there should be no need for a collaps. Treating the interaction non-unitarily introduces an artificial split, either a contradiction or incompleteness

 

[Fredrik]

That's not consistent.

Decoherence shows that even the interaction with the environment can be treated according to QM, therefore there should be no need for a collaps. Treating the interaction non-unitarily introduces an artificial split, either a contradiction or incompleteness

I know, but I didn't say that we should treat it non-unitarily. If if seemed that I did because I was saying what I said in a thread where other people have mentioned such things, then I guess I should have been more specific.

In principle, one can imagine a system S that's an isolated subsystem of an isolated system S'. In that case, S evolves according to the Schrödinger equation until its interaction with the other parts of S' become non-negligible. After that point, S' still evolves according to the Schrödinger equation until its interactions with its environment become non-negligible. So for a while, in this highly idealized scenario, S fails to satisfy the Schrödinger equation, but must still behave in a way that's consistent with the fact that S' does.

This is of course assuming that the Schrödinger equation still holds for larger, more complicated systems, or at least that it would hold if we could isolate those systems from their environments. I don't think that this must necessarily be the case, but I also don't have a reason to think that it's not, other than the fact that all other theories we have found so far have a limited range of validity.

 

[tom.stoer]

I understand; for your scenario QM still makes perfectly sense