This is fundamental equation of Quantum mechanics. It comes in two versions: the time-independent version, and the more general but less useful time-dependent version.
Theoretically, everything in our everyday world follows this equation (the time dependent one). This includes all of chemistry, biology, anthropology, geology, economics, etc. While this equation is probabilistic, on most large scales it quickly becomes effectively deterministic at it's behavior merges with classical mechanics. So if one could solve my Schrodinger equation and perhaps the equation for my local area they would probably be able to predict a great deal of my future.
Unfortunately, or perhaps fortunately, while this may be true, it is impossible. To try and manipulate this equation with anything much more complicated then a few atoms, or perhaps an infinite lattice of identical atoms, and you have a nightmare of a problem. You run into the same many body problem people have when trying to solve any system with too many degrees of freedom plus a big, ugly algebra mess.
Hence, our non-deterministic Romantic world is preserved. We don't have to worry about mad scientists using Quantum mechanics to play God.
Note: The only things that violates this equation are things approaching the speed of light and perhaps things at extreme energies/times/masses/temperatures. Not something I see every day.