It's been a long time since my QM course, but, assuming by unitary you mean deterministic, then the answer is that time evolution of a particle's state is NOT deterministic. It is probabilistic. For large collections of particles, the probabilities come to overwhelmingly favor a single average state, but the state of any single particle remains non-deterministic.

This is in contrast to, for example, classical statistical mechanics where individual particles are considered to behave deterministically, but we simply lack the ability to measure their behavior. In (the Copenhagen interpretation of) QM there is no underlying determinism at all.

You might not like this idea (it's certainly non-intuitive), but decades of experiments have shown that the alternatives are all, most likely, wrong.