time evolution

Time evolution is how a quantum state changes from one moment to the next. Given the state of a system right now, the Schrödinger equation tells you, with no ambiguity, what the state will be a second later, an hour later, or a century later. In this respect quantum mechanics is strictly deterministic: the wavefunction marches forward like clockwork, set entirely by the system's energy.

It is worth being careful here, because quantum mechanics has a reputation for randomness. The randomness shows up only when you make a measurement and read off a definite outcome. Between measurements, while the system is left alone, nothing is random at all — the state simply flows smoothly and predictably according to the equation. The two pieces, smooth evolution and abrupt measurement, are very different kinds of change.

The engine driving this flow is the Hamiltonian, the operator that encodes the system's energy. If you know the Hamiltonian, you know the future of the state, in principle exactly. Because energy is conserved for an isolated system, this lawful unfolding also keeps the total probability fixed at one, so the state never gains or loses 'reality' as it evolves.