Time-Evolution Operator

Canonical treatment: Time-Evolution Operator maintains the spectral forms, time ordering, worked examples, exercises, and references.

This bridge is intentionally limited to the definition and properties needed in Core Formalism.

Definition and core properties

The propagator maps initial states to later states:

$$\lvert\psi(t)\rangle=U(t,t_0)\lvert\psi(t_0)\rangle, \qquad U(t_0,t_0)=I.$$

For a closed system it is unitary and composes as

$$U(t_2,t_0)=U(t_2,t_1)U(t_1,t_0), \qquad U(t,t_0)^{-1}=U(t_0,t)=U(t,t_0)^\dagger.$$

For a time-independent self-adjoint Hamiltonian,

$$U(t,t_0)=\exp\!\left[-\frac{i}{\hbar}H(t-t_0)\right].$$

The operator evolves density matrices by $\rho(t)=U\rho(t_0)U^\dagger$. Continue to the canonical page for differential equations, time-dependent generators, spectral decompositions, and transition-amplitude calculations.