Such parametrization is smooth , which means that unitary matrices form a manifold. So unitary matrices form a Lie group. Such operators form a Lie group. An infinitesimal evolution of a quantum state in Hilbert space is accomplished by a generator of the unitary group. That generator is called a Hamiltonian matrix.
Hamiltonian matrices belong to the Lie algebra of the Lie group of unitary transformations.
These statements aren't really as profound as they may appear to be. All they express is the fact that is a probability that a quantum particle that is in state can be found in that state, and that probability is, of course, 1. And, that, as we evolve the system, its quantum state changes to some other quantum state, but the same tautology still applies to it. Quantum mechanics would become much more interesting if some other, possibly non-probabilistic interpretation could be given to . We would no longer be restricted to a sphere of radius 1 then, and evolution operators would no longer have to be unitary.