Why is wave function collapse mysterious? There are lots of questions and answers on this site about wave function collapse (for example, How does a Wavefunction collapse?, Why does a wavefunction collapse when observation takes place?, How does wave function collapse when I measure position?, and others).  However, after reading a few of them, I still don't understand why this is such a difficult topic.  Since I don't know very much about quantum mechanics, I'm probably just missing a fundamental concept somewhere; I'm trying to figure out what it is.
As I understand it, in quantum mechanics, everything that exists is described by a wave function.  A particle in a box has a wave function.  Light in the double slit experiment has a wave function.  I have a wave function, albeit an extremely complex one.
When two particles interact, one can use quantum field theory to write a wave function describing the system before and after the interaction.  For example, if a free electron approaches a proton, one can write a wave function describing the time evolution of the system.  At any given time in the future, there is a probability that the electron will be near the proton with a low energy, indicating that it has emitted a photon and has been captured into an atomic orbital.  There is also a probability that it will be far away with a high energy, indicating that it remains free.
It seems to me that when we make a "measurement" of a quantum mechanical system, exactly the same thing happens.  One particle interacts with another group of particles (the observer), and in principle, one could write a wave function describing the time evolution of the whole system.
For example, a physicist might measure the spin of an electron.  After the measurement is made, there is some probability that the electron is spin-up and the atoms in the physicist's brain rearrange themselves in such a way that the physicist believes the electron is spin-up.  There is a probability that the electron is spin-down and the physicist believes it is spin-down.  There is also a probability that the physicist makes an error, a (tiny) probability that the physicist is actually in the Andromeda Galaxy and didn't make the measurement at all, and so on.
Indeed, the entire idea of the physicist "really" being in the Andromeda Galaxy is perhaps not very meaningful.  It would be more precise to say that the physicist is really a giant wave function with a tiny amplitude a few million light years from the particle being measured.
Since everything is a wave function, both before and after the measurement, this explanation sidesteps the entire idea of the particle's wave function "collapsing."  Instead, the particle is just interacting with other particles.
Is this a valid way of thinking about wave function collapse?  If so, what is missing?  Why is the interpretation of quantum mechanics considered to be an interesting question?
 A: Yes what you describe is a valid way of describe the wave function collapsing. On your account there is a universal wavefunction and it, at all times, evolves unitarily. There is in fact no collapse. Wavefunction collapse is an explicitly non-unitary time evolution of whatever wavefunction is under consideration and that is what is sometimes controversial. Schrodinger's equation tells us that systems evolve unitarily and wavefunction collapse tells us that systems evolve non-unitarily. One major criticism of the Copenhagen interpretation is that it doesn't clearly explain when the universe follows the unitary evolution and when it follows non-unitary evolution. That makes it an incomplete theory at best.
Anyways, what you describe --- only unitary evolution --- often goes under the name the Many Worlds Interpretation of quantum mechanics. However I think this is a pretty bad misnomer as it gets pretty scifi pretty quick and people start talking about divergent universes etc. when what is going on is really just as you describe, the universe is just undergoing unitary evolution which leads to entanglement and superposition states.
What is the problem with this approach? In an objective world there is no problem. However, as far as we* know the universe has subjective components as well. For example, I have a feeling or experience or qualia as to what it feels like to observe an electron in the spin up state (perhaps I have an apparatus that lights up a different colored light depending on the spin state measured by some spin measuring apparatus). But I do NOT have a feeling or experience of ever being in a superposition of having seen both spin up and spin down.
However on your account, the particles that make up my body, my brain etc. are in a superposition of having experienced both of those things.
I think most modern physicists probably harbor a tacit dualist perspective on the mind-body problem and would subscribe to the idea that our subjective thoughts are correlated with the physical state of our brains and bodies. In fact, we might go so far as to suppose that mental states a one to one with physical states.
Your interpretation is not compatible with this naive dualist perspective. On your interpretation a person's body would be in a superposition of having experienced both a spin up and spin down electron. What mental state would they be in? You could say it is random at the end of the measurement. Say the person's mind experiences spin up right after the measurement. Ok. But what about 10 seconds after the measurement? On your account the wavefunction should still be a superposition of the person's body having seen up and down. So do the dice get rolled again to determine what the person experiences in this new instant?
Is it random from instant to instant which experience we have? That is a bit solipsistic.
Is there a rule that says if your mental state experiences spin down at the end of the experiment it will also experience spin down 10 seconds later despite the wavefunction having equal weight for both probabilities? If so our theory should probably be able to describe that rule.
Or is it somehow possible for a person to have multiple simultaneous, and contradictory experiences? This has implications for what is meant by one's personal identity.
What your, and the many worlds interpretation, fatally fails to do is provide any account whatsoever for how physical states are correlated with subjective experience. This comes for free in classical physical theories so we typically don't think of this as being a desiderata for physical theories. This comes for free in classical theories because we can say the E&M fields which hit our eyeballs move charges in our optical nerves which affect the neurons in our brain, and because our mental states are correlated with the physical state of our bodies (possibly in a 1:1 way) it is clear that measurement results should cause us to experience particular things. It is not so clear quantum mechanically however. What the copenhagen interpretation, or spontaneous collapse does is basically brutally jam this correlation between mental and physical states back into the theory by hand by demanding that the system collapses into one state or the other so that we avoid the conundrum of people having manifold simultaneous experiences.
In any case, there are many philosophical issues here that I won't be able to present in a very coherent way but I did want to share some of my thoughts and some references.
See

*

*Maudlin, T. Three measurement problems. Topoi 14, 7–15 (1995). for a great introduction to the quantum measurement problem.


*Decoherence and the Quantum to Classical Transition by Schlosshauer. A great intro to decoherence that can help you avoid some traps that come with thinking about decoherence in the context of unitary evolution and the many worlds interpretation generally.


*https://www.quantamagazine.org/why-the-many-worlds-interpretation-of-quantum-mechanics-has-many-problems-20181018/
*Or at least I
A: The mysterious aspects of quantum mechanics are both exaggerated and confused by popular science writers.
Many of the quantum effects have become targets for hype in part owing to the original Copenhagen interpretation which emphasised the role of measurement in a way that allowed it to take on an undue significance. A measurement, after all, is simply an interaction between the quantum system being observed and the quantum systems that form the detecting part of the measuring apparatus. When we say that a particle's wave function collapses on measurement, we really mean that it changes when the particle interacts with other collections of particles, which sounds far less mysterious.
All sorts of nonsensical ideas have developed as a result of undue significance being attached to measurement, including bizarre suggestions that consciousness causes collapse, and the endless misguided hype of Schrodinger's cat.
What one should bear in mind is that quantum effects occur whenever particles interact in nature, regardless of whether measurements are being made. One should also remember that QM is a mathematical model used to make calculations, and there is a danger of interpreting it too literally. Take the case of a vibrating guitar string: its motion can be expressed mathematically as a superposition of normal modes of vibration; do we take that to mean that it is really an infinite collection of vibrating states? No, we understand it to mean only that the motion can be represented that way mathematically.
That all said, there is genuinely still a mystery at the heart of quantum mechanics, and none of the so-called interpretations can, yet, satisfactorily account for all of the observed effects.
A: This is because the collapse of quantum mechanical waves is only a qualitative theory. Quantum mechanics doesn't really tell you what it means. Imagine that a light source sends out a wave, which shoots in all directions. How can these waves suddenly converge from all directions to a point on the measurement screen. The position of this point is still random. This point is an absorber. A photon is obtained at the position of the absorber. The problem is that this wave may have reached the boundary of the universe. How does this wave converge from all directions to a point in infinite space? Collapse should be a physical process. Quantum mechanics doesn't tell us the equation that this process satisfies. Quantum mechanics, especially the interpretation of the so-called Copenhagen school, tells us that waves are probability waves. Probability can collapse. All these are against our common sense.
Why does quantum mechanics introduce wave collapse? Because we measure particles one by one. This particle is point-to-point propagation energy. However, the particle has the properties of wave, such as the interfere when the particle passes through a double slit. These interference must be described by the wave equation. But the wave is to defuse the whole space, and to be transmitted outside the universe. This is completely different from particles. Particles are point-to-point propagation. Besides, this interpretation of the Copenhagen school was not endorsed by Einstein and Schrodinger. Einstein opposed this interpretation with the famous saying that God would not roll the dice. Schrodinger used the so-called Schrodinger cat theory to resist the interpretation of the Copenhagen school. So what is collapse? In fact, this problem has not been really solved. It is precisely because this problem has not been solved, we have different opinions on it, so this problem is very interesting.
For example, John Cramer's transactional interpretation of quantum. There is the concept of continuous collapse in the transactional interpretation. The transactional interpretation agrees with the advanced wave theory in Wheeler Feynman's absorber theory. The radiator produces a retarded wave. The absorber generates an advanced wave. The advanced wave should shake hands with the retarded wave. Transactional interpretation also has a lot of support in quantum mechanics, and John Cramer's paper has been cited by 800 people. However, John Cramer didn't tell us what the handshake in the transactional interpretation was. Recently, there is a very new theory: mutual energy theory, which seems to give a more specific description of this handshake. According to the mutual energy theory, if the retarded wave and the advanced wave are synchronized, there will be mutual energy flow. The process of generating mutual energy flow is the handshake between retarded wave and advanced wave. The mutual energy theory also supports the existence of the advanced waves. However, the waves in the mutual energy theory not only have retarded waves, advanced waves, but also two corresponding time reversal waves. There are four waves in total. The energy flow of the retarded wave emitted by the radiation source is offset by the time reversal wave corresponding to the retarded wave. The energy flow of the advanced wave emitted by the sink (light absorber) is offset by the time reversal wave corresponding to the advanced wave. This cancellation process can be seen as the reverse collapse of the wave. Reverse collapse is different from wave collapse. Wave collapse is the collapse at the destination of the wave. However, wave reverse collapse + mutual energy flow can play the role of wave collapse. Therefore, mutual energy theory actually gives a very specific mathematical description of wave collapse for the first time. In mutual energy theory, mutual energy flow is equivalent to photon. Therefore, according to the mutual energy theory, the collapse of the wave emitted by the light source is actually the first reverse collapse to the light source, and then the light source emits a photon to the light sink at the same time. Photons are mutual energy flow, which is the interaction of retarded wave and advanced wave (handshake).
Mutual energy theory includes three axioms: mutual energy principle, self energy principle and energy conservation law (Not the Poynting theorem, but a generalization of the mutual energy theorem); Mutual energy theorem, mutual energy flow theorem, advanced wave existence theorem, radiation does not overflow universe theorem (Law), and the interpretation of mutual energy flow in quantum mechanics. Mutual energy theory makes up for the loopholes in the classical electromagnetic field theory. For example, in the classical electromagnetic field theory, electromagnetic waves do not collapse. But if the classical electromagnetic wave does not collapse, it will certainly overflow our universe, which is obviously wrong. According to the theory of mutual energy, the wave collapses inversely. That is, all the energy of the wave returns to the light source. How does electromagnetic energy spread? Through mutual energy flow. The mutual energy flow is generated by the interaction between the retarded wave emitted by the light source and the advanced wave emitted by the light sink. This mutual energy flow is photons. Because the mutual energy flow is composed of retarded wave and advanced wave, the mutual energy flow is photons, so photons have many wave properties. If you want to know more about mutual energy theory, search for "mutual energy flow" and "mutual energy principle".
A: The problem of measurement is incisively approached by the Epr paradox. In there there is a bipartite system, from which it is said that out of measurement on one subsystem without disturbing the second predictions could be made on the second :
System : $|\psi\rangle=(\pmatrix{1\\0}\pmatrix{0\\1}-\pmatrix{0\\1}\pmatrix{1\\0})/\sqrt{2}$
Observable in A : $\pmatrix{\cos a&\sin a\\ \sin a&-\cos a}$
Now as measurement one could seek an operator that locally applied on the state disentangles it. It is easy to notice that such an operator is irreversible, ie a matrix with 0 determinant. Qm tells the endstate in A is an eigenvector $u$ of the observable in A.
Hence as a many-to-one irreversible evolution operator for measurement one could take the matrix having as columns the eigenvector u. Then this matrix is applied on the state.
However this gives that measuring A and B in the system gives as result the impossible event.
Thus one should maybe consider the column of the matrix as being multiples of u to avoid the problem.
