Quantum dynamics

Quantum dynamics describes the time evolution of a quantum state and its observable predictions. For an isolated system with Hamiltonian operator $H$, the Schrödinger equation is $i\hslash {\partial }_t|\psi ⟩=H|\psi ⟩$. Equivalently, a density matrix satisfies $i\hslash \dot{\rho }=[H,\rho ]$.

For time-independent $H$, the evolution operator is $U(t)=e^{-iHt/\hslash }$. An energy eigenstate changes only by a global phase, whereas a superposition develops relative phases and can display oscillating observables. Interaction with an unobserved environment generally requires open quantum system dynamics rather than a unitary evolution for the subsystem alone.