And that state is a completely made up mathematical artifact. The wavefunction is not measurable, but its absolute value squared is. This does not mean it's no

And that state is a completely made up mathematical artifact. The wavefunction is not measurable, but its absolute value squared is. This does not mean it's not "real". I understand what you mean, but you need to provide us with another better answer that supports all observations before saying it's "completely made up". 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.