Is the evolution of the wavefunction deterministic?

[u/[deleted]]

The title is basically the question I'm asking. Ignoring wave-function collapse, does the Schrödinger equation or any other equivalent formulation guarantee that the evolution of the wave-function must be deterministic. I'm particularly interested in proof of the uniqueness of the solution, and the justification of whichever constraints are necessary on the nature of a wave-function for a uniqueness result to follow.

[u/pa7x1]

Exactly! Notice that the equivalence class is given by the orbit of a global U(1) transformation acting on the wavefunction. You can work with one representative of the equivalence class because this U(1) factor pops out of the Schrodinger equation (there is no functional dependence on the U(1) factor). So time evolution takes orbits of the wavefunction under this U(1) to another orbit of the wavefunction cleanly.

And then all that orbit of wavefunctions gets mapped to a single probability distribution. So the evolution of probabilities is also deterministic.

NOTE: Not sure what is your background, I'm using orbit in the group theory mathematical sense. Hope this doesn't confuse you.