time-evolution operator

The time-evolution operator, usually written U(t), is the single object that takes the state of a system at the start and hands you the state at a later time. Feed it the initial state and it returns the evolved one: it is the machine that performs the marching-forward described by the Schrödinger equation, packaged as one ready-to-apply rule rather than a differential equation to solve afresh each time.

When the system's energy does not change with time, U(t) has a beautifully compact form: it is the exponential of the Hamiltonian times the elapsed time, divided by Planck's reduced constant. That compactness hides real depth — the Hamiltonian, the operator of energy, literally generates the flow of time for the state, in the same way a velocity generates motion along a path.

Crucially, U(t) is unitary, which is what guarantees that probability is conserved as the state moves. It also chains together sensibly: evolving for two seconds then three more is the same as evolving for five, and you can always undo an evolution by running the inverse operator. These simple properties make U(t) the cleanest way to think about quantum dynamics.