A _density matrix_, or a _density_ operator is a representation of a [quantum system](Quantum%20systems.md) constructed from either one or more than one [state vector.](State%20vector) This is such that a [pure state density matrix](Pure%20state.md#Density%20Matrix) is written as an [outer product](Outer%20product.md) of itself and [mixed state](mixed%20state.md#Density%20Matrix) density matrix is written as a probability weighted sum of outer products of state vectors. This means that in general, given a density matrix $\hat{\rho},$ [$\hat{\rho} = \sum_i^n p_i|\psi_i\rangle\langle\psi_i|.$](mixed%20state.md#^8a9096) ^bab548 For a pure state, the density matrix reduces to [$\hat{\rho} =|\psi\rangle\langle\psi|.$](Pure%20state.md#^98c745) Any quantum system may be described in terms of a density matrix as long as we define its corresponding [state vectors](State%20vector.md) to be [a Hilbert Space in Quantum Mechanics](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md). However, the motivation for describing quantum systems with density matrices arises from the need to be able to apply operators to systems where more than one state might be present, such as in a [mixed state.](mixed%20state.md) Density matrices give us a framework to do this in an equivalent and consistent manner for both mixed states and [pure states.](Pure%20state.md) Thus when considering an _[ensemble](Ensembles%20of%20quantum%20systems.md)_ of quantum mechanical systems we will tend to want to model systems with [density matrices.](density%20matrix.md) ^ea24b6 %%whether ANY quantum systems can be written as a density matrix is an open question and tied also to the way we define Hilbert spaces or rigged hilbert spaces in quantum mechanics. How does this picture fit with Rigged Hilbert Space? this discussion here is inspired by talking to Rudolf Gross at University of Cologne on the topic.%% # Properties of density matrices All density matrices have the following properties: 1) $\hat{\rho}^{\dagger}=\hat{\rho}$ ([Hermiticity](Hermitian%20operators.md)) ^450e45 2) $\hat{\rho}$ is a [positive semi-definite matrix](Positive%20semidefinite%20operators.md#Positive%20semi-definite%20matrices) 3) [tr](Trace.md)$(\hat{\rho})=1$ 4) $\mathrm{tr}(\hat{\rho}^2)\leq 1$ Property [1.](density%20matrix.md#^450e45) follows for [pure states,](Pure%20state.md#Density%20Matrix) since for a pure state [density matrix,](density%20matrix.md) [$\hat{\rho} =|\psi\rangle\langle\psi|$,](Pure%20state.md#^98c745) $(|\psi\rangle\langle\psi|)^\dagger=|\psi\rangle\langle\psi|$ and follows more generally for [mixed states,](mixed%20state.md#Density%20Matrix) [$\hat{\rho} = \sum_i^n p_i|\psi_i\rangle\langle\psi_i|$,](mixed%20state.md#^8a9096) since $(p_i|\psi_i\rangle\langle\psi_i|)^\dagger=p_i|\psi_i\rangle\langle\psi_i|$ (noting that $p_i\in\mathbb{R}$), and [$(A+B)^{\dagger}=(A^\dagger+B^\dagger)$.](Adjoint.md#^6de25c) ## [pure state properties](Pure%20state.md#Properties) Density matrices that model [pure states](Pure%20state.md) have the following distinguishing properties:   ## [mixed state properties](mixed%20state.md#Properties) Density matrices that model [mixed states](mixed%20state.md) have the following distinguishing properties:   # Density matrices for systems with continuous spectra %%Pg. 86 of Sakurai%% # Measurements on density matrices ## Expectation value of a density matrix  ([... see more](expectation%20value.md#Density%20matrix%20expectation%20value)) ### Ensemble averages  # Unitary transformations on density matrices  ## Dynamics with density matrices The dynamics of a [closed quantum system](Closed%20quantum%20systems.md) described with a [density matrix](density%20matrix.md) are modeled by the [von Neumann Equation,](von%20Neumann%20Equation.md)[$\frac{\partial\hat{\rho}}{\partial t} = -\frac{i}{\hbar}[\hat{H},\hat{\rho}].$](von%20Neumann%20Equation#^b69229) # Reduced density matrices ([... see more](Reduced%20density%20matrices)) # Modeling thermodynamic quantum systems with density matrices #QuantumMechanics/QuantumStateRepresentations/DensityMatrices #QuantumMechanics/MultiParticleQuantumSystems