The density matrix or density operator is an alternate representation of the state of a quantum system for which we have previously used the wavefunction. Although describing a quantum system with the density matrix is equivalent to using the wavefunction, one gains significant practical advantages using the density matrix for certain time-dependent problems—particularly relaxation and nonlinear spectroscopy in the condensed phase. 5.1: Introduction to the Density Matrix The density matrix is defined as the outer product of the wavefunction with its conjugate. Its name derives from the observation that it plays the quantum role of a probability density. 5.2: Time-Evolution of the Density Matrix The equation of motion for the density matrix follows naturally from its definition and the time-dependent Schrödinger equation. 5.3: The Density Matrix in the Interaction Picture For the case in which we wish to describe a material Hamiltonian under the influence of an external potential, we can also formulate the density operator in the interaction picture. Thumbnail: measured density matrix of a thermal state. (CC SA-BY 3.0 unported; measured density matrix of a thermal state).