# 5: The Density Matrix

- Page ID
- 107243

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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).