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?