In physics, quantum dynamics is the quantum version of classical dynamics. Quantum dynamics deals with the motions, and energy and momentum exchanges of systems whose behavior is governed by the laws of quantum mechanics. Quantum dynamics is relevant for burgeoning fields, such as quantum computing and atomic optics.

In mathematics, quantum dynamics is the study of the mathematics behind quantum mechanics. Specifically, as a study of dynamics, this field investigates how quantum mechanical observables change over time. Most fundamentally, this involves the study of one-parameter automorphisms of the algebra of all bounded operators on the Hilbert space of observables (which are self-adjoint operators). These dynamics were understood as early as the 1930s, after Wigner, Stone, Hahn and Hellinger worked in the field. Mathematicians in the field have also studied irreversible quantum mechanical systems on von Neumann algebras.

The dynamics of a quantum system are governed by a specific equation of motion that depends on whether the system is considered closed (isolated from its environment) or open (coupled to an environment).

A closed quantum system is one that is perfectly isolated from any external influence. The time evolution of such a system is described as unitary, which means that the total probability is conserved and the process is, in principle, reversible. The dynamics of closed systems are described by two equivalent, fundamental equations.

The most common formulation of quantum dynamics is the time-dependent Schrödinger equation. It describes the evolution of the system's state vector, denoted as a ket ${\displaystyle |\psi (t)\rangle }$. The equation is given by:

$${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }$$

Here, ${\displaystyle i}$ is the imaginary unit, ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle |\psi (t)\rangle }$ is the state of the system at time ${\displaystyle t}$, and ${\displaystyle {\hat {H}}}$ is the Hamiltonian operator—the observable corresponding to the total energy of the system.

The Schrödinger equation is powerful but applies only to pure states. A more general description of a quantum system is the density matrix (or density operator), denoted ${\displaystyle \rho }$, which can represent both pure states and mixed states (statistical ensembles of quantum states). The time evolution of the density matrix is governed by the Liouville-von Neumann equation: