It really goes deeper than just a theoretical demand on a particular domain. The Hamiltonian for any system must be unitary, because that preserves the total probability at one.
This is important because if I start with some state and let it evolve for a while the system must afterwards exist in some state which means that the sum of the probabilities taken across all final states must come to 1. Otherwise things can undergo---in the words of Douglas Adams---"a sudden and gratuitous total existence failure".
Nor is it acceptable to start with a single state and end up with the probability to exist in one of all possible state larger than one. What would that even mean? Sudden and gratuitous extra existence?
This was probably mentioned on the first day you started studying quantum mechanics, but it is so obvious that student often don't take much note of it.
This post imported from StackExchange Physics at 2014-04-13 14:45 (UCT), posted by SE-user dmckee