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The collapse of the wavefunction by comparing it with the Schrodinger equations has some differences: it is higly non-linear while the Schrodinger equation is linear, it is non-local as proven by Bell's theorem while the Schrodinger equation is local (thats why we can combine it with SR to derive QFTs), it is an irreversible process while the Schrodinger equation is time reversible so I dont think they must be part of the same theory, they are just too different.

So why the need to include the wavefunction collapse as part of the quantum theory?

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    $\begingroup$ Schroedinger's equation is a non-relativistic description, so it is not really local. A packet localized to a finite region spreads and has some intensity in infinity immediately, there is no step ("shock wave") propagating with finite speed. $\endgroup$ Commented Aug 7 at 13:58
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    $\begingroup$ Collapse is a component of some interpretations of quantum physics. $\endgroup$ Commented Aug 7 at 17:16
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    $\begingroup$ @my2cts OP said SE is local, I don't think that is a good description, so I've pointed that out. $\endgroup$ Commented Aug 10 at 14:18
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    $\begingroup$ @my2cts I don't see your point, SE has no $c$. Local theory should respect special relativity, finite speed of propagation of interaction. Non-relativistic classical mechanics equations of motion and SE for two or more particles do not respect it, thus these are not local. $\endgroup$ Commented Aug 10 at 14:52
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    $\begingroup$ @RootGroves why do you think wavefunction moves at a finite speed? In spectral resolution of a localized wave packet, we can have as high momenta as we want, and this is non-relativistic theory, where group velocity $v = \frac{d\omega}{dk}=\frac{dE}{dp}= p/m$, so speed isn't capped. $\endgroup$ Commented Aug 10 at 18:36

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The expression "wavefunction collapse" is connected to the fact that in QM the wavefunction describes only probabilities for the possible outcomes of the measurement of an observable and that at the measurement the wavefunction is changed to the eigenfunction corresponding to the specific observed eigenvalue of the observable. An immediately repeated measurement then again yields the same eigenvalue. This non-linear "wavefunction collapse" is not described by the unitary time development of the solution of the Schrödinger equation and is usually added as an additional feature (axiom) to QM in order to describe the outcome of measurements.

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    $\begingroup$ Well ok then what counts as a observer?Why does the state of a observarble changes instanteneously from a matrix with generally different than 0 elements to a matrix with 1 element 1 and the other 0?\ $\endgroup$ Commented Aug 9 at 13:18
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    $\begingroup$ @Root Groves - These questions seem to go beyond your original question. What counts as an "observer" is actually a tricky question in QM. And if I understand you correctly, your second sentence refers to changing matrix elements. I can only guess that you might have a density matrix in mind. It would probably be better if you pose a new question where you describe clearly what you want to know. $\endgroup$ Commented Aug 9 at 19:01
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In classical physics the evolution of a physical quantity such as the $x$ position of a particle is described by a function $x(t)$ such that if you measure $x$ at time $t$ you will get the result $x(t)$.

In quantum theory, the evolution of a measurable quantity is described by an observable whose value at any given time is a linear operator. The eigenvalues of that operator are the possible results of measuring that quantity and quantum theory predicts the probability of each possible value. In general the probabilities depend on all what happens to all of the possible values. This is called quantum interference. For an example see Section 2 of this paper

https://arxiv.org/abs/math/9911150

If you pick up a ball and throw it through a nearby doorway, you will only see it go along one of the possible routes through the door and its trajectory will not depend on what is happening on other possible routes. This appears inconsistent with quantum theory.

According to the equations of motion of quantum theory, when you copy information out of a quantum system, interference is suppressed, this is called decoherence

https://arxiv.org/abs/1911.06282

This effect has been experimentally tested on many systems including large molecules and superconductors. For objects evolving on scales of space and time you can see in everyday life, information is being copied out of them on much smaller scales of space and time. As a result, decoherence suppresses interference for those systems very effectively. Decoherence doesn't eliminate the other possible states of such systems, but it does prevent them from interfering and on the scales of everyday life it predicts that they evolve according to the equations of classical physics to a good approximation. On those scales decoherence sorts different versions of systems into layers each of which approximately resembles the universe as described by classical physics. This is often called the many worlds interpretation but it is just a consequence of treating quantum theory as one would treat any other physical theory:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

Collapse is a process that is alleged to eliminate all of the possible values of a measurable quantity except for one. This is incompatible with the equations of motion of quantum theory and is not the same as decoherence. In many textbooks collapse is invoked without any explanation of how it happens or what its consequences might be. Some physicists have come up with variants of quantum theory with different equations of motion that include collapse, see

https://arxiv.org/abs/2310.14969

It should be noted that if such theories are to agree with experimentally tested predictions of quantum theory they must be both non-local and non-Lorentz-invariant

https://arxiv.org/abs/1808.04966

As such collapse theories are incompatible with quantum field theories and don't currently reproduce their tested predictions:

https://arxiv.org/abs/2205.00568

It should also be noted that collapse is incompatible with a lot of quantum measurements used in labs such as unsharp and repeated measurements:

https://arxiv.org/abs/1604.05973

AFAICT the motivation for collapse is a philosophical ick that has nothing to do with understanding how the world works, how experiments work or predicting experimental results but you should read the literature and check for yourself.

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Quantum theory, which includes quantum mechanics, quantum field theory, quantum optics, etc., form part of theoretical physics. Physics is a science that follows the scientific method. The theoretical part is combined with the experimental part to develop an understanding of the physical world.

The notion of the collapse of the wave function falls under the topic of interpretations of quantum mechanics, of which there are many. None of them have ever been ruled out because they are not assessable with the scientific method. Therefore, these interpretations do not form part of science and is therefore not strictly speaking part of physics. One can place them under philosophy of science, perhaps.

So, although the collapse of the wave function is an idea inspired by quantum theory, it is not strictly speaking part of it.

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    $\begingroup$ Different "interpretations of quantum mechanics" are really slightly different theories, and some differences are experimentally falsifiable. Orthodox quantum theory uses quantum state to describe a single system, not just an ensemble, and thus has collapse. This can be empirically confirmed by subsequent measurements on single system. E.g., an SG-magnet splits a beam of atoms into two directions; an atom in one chosen direction has a collapsed quantum state consistent with that direction, and this can be confirmed by a subsequent measurement of the same atom. $\endgroup$ Commented Aug 8 at 11:28
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    $\begingroup$ Yes, that's why I wrote some differences in "interpretations" (really, different theories) are experimentally falsifiable. MWI's many worlds is not experimentally falsifiable, because whatever happens in a single experiment, those many worlds as a whole can still be consistent with the Schroedinger equation ( we have no way of knowing what happens to those other worlds). $\endgroup$ Commented Aug 8 at 11:34
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    $\begingroup$ When the atom is prepared in a definite spin state (involving collapse of its previous spin state), MWI would describe this by saying quantum state describing all the worlds did not collapse, even though the single system behaves as with collapsed state. While MWI does not have collapse of the many-world quantum state, it has to respect the fact that in the branch describing the experiment succession, the single atom quantum state is collapsed. $\endgroup$ Commented Aug 8 at 11:49
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    $\begingroup$ An interpretation is alway compatible with the predictions of quantum mechanics and is not a version of quantum theory. The question is if 'collapse' is falsifiable. If it is, and if is not falsified, then it can be part of quantum theory. If it is not falsifiable, then it is not part of quantum theory but of an interpretation of it. As far as I know 'collapse' is not falsifiable. $\endgroup$ Commented Aug 10 at 5:16
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    $\begingroup$ @my2cts when we do an experiment on a single atom, we use $\psi$ to describe this single atom. You may introduce ensemble as a mental device to think about probabilities. After a measurement result is created, the psi has to change to be in agreement with this result. You may introduce another ensemble with different probabilities of results. Whatever we say, we can check that the appropriate $\psi$ has changed after the measurement. $\endgroup$ Commented Aug 10 at 17:56

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