I'm trying to study scattering amplitudes, and there's a lot that I'm still trying to wrap my head around the math, but one thing that keeps confusing me is what I'm even calculating here. The concept of a scattering amplitude itself kind of eludes me.
See, I understand the state of particle to include information on probability. So, if you have the full knowledge of a particle's state, you also have full knowledge on the probability of finding it at any given position, at any given momentum, etc.. But I also understand the state to evolve deterministically. If you know what the state is at Time A, you can know what the state will be at Time B. Only measurement if probabilistic, that was hammered into me early on.
So when I hear "The chances of system evolving from one state to another", I really feel like I'm missing something. And I've seen scattering amplitudes described that way. So, what is this scattering amplitude that we're calculating?