Measurement effects are not captured by any QM equation.
The equations are all time symmetric, yes. But the theory is not.
The equations are all time symmetric, yes. But the theory is not.
You might argue that the same is true for the collapse postulate and the Born rule but I have to disagree. Yes, it gives the correct answer in as many experiments, but those are parts of the experiments that we don't really understand. We have classical system interact with quantum systems and our measurement devices are just such gigantic quantum systems that we can't treat them quantum mechanically.
The collapse postulate is obviously true in a kind of classical limit, but it is really just an emergent law and nothing fundamental. If you could show that a wave function collapse can occurs in quantum system before we measure it, that would change the game. But that is not the case, the wave function collapse only occurs when we throw a gigantic classical system at the quantum system which we can not treat properly due to its complexity.
So I argue people are rightly ignoring wave function collapse when they talk about time symmetry because it is just an emergent phenomenon. And there are several ideas that can explain why it looks like a wave function collapse occurs, so we are not simply throwing a corner stone of our theory out of the window.
I would love to find out the answer to this! There needs to be some more surveys of physicists...
> And there are several ideas that can explain why it looks like a wave function collapse occurs
I don't think it is at all as simple as you are implying. This is still very much an open question, even if most physicists don't care to think about it.
Given that a reversible linear mechanics is so much more useful than one that isn't, I think it's way more likely that the universe isn't in fact linear, but the best way people got to mapping it was to shove all the non-linearity at the border of the mechanics, right before they get something from the experiment.
The result of a measurement is not dealt with by the equation. Schrödinger only lets you calculate the probabilities of measurement outcomes, but not which one it will be. That's why Einstein proposed we just didn't knew the whole picture. He thought we needed more information.
On the case of QM, that's the set of equations, observable operators, and the instructions of hoe to interpret them. None of those predict anything alone.
Besides, the Schrodinger equation does not predict how the state will evolve after a measurement is done. That's the entire problem.
And that state is a completely made up mathematical artifact.
And yes, the Schrödinger equation predicts exactly how a state evolves, even after a measurement. It does not tell you what the outcome of a measurement is - you give it the outcome, and it gives you the time evolution.
There is a problem, and it lies in what happens during measurement - or what is the exact nature of measurement. But given any state, the SE tells you how it evolves.