The von-Neumann equation is a differential equation modeling [time evolution](Equations%20of%20motion%20in%20quantum%20mechanics.md) for [ensembles of quantum systems,](Ensembles%20of%20quantum%20systems.md) i.e. systems modeled with [density matrices,](density%20matrix.md) $\hat{\rho}$. It is written as $\frac{\partial\hat{\rho}(t)}{\partial t} = -\frac{i}{\hbar}[\hat{H},\hat{\rho}(t)].$ where $[\hat{H},\hat{\rho}]$ is a [commutator](Commutators%20in%20quantum%20mechanics.md#Commutation%20relations%20with%20density%20matrices) contain the density matrix and a [Hamiltonian](Hamiltonian%20operator.md) describing the system. ^b69229 Following from the [correspondence between Poisson brackets in classical physics and commutators in quantum mechanics,](Quantization.md#^c2dd51) one finds that the [von Neumann Equation](von%20Neumann%20Equation.md) as quantum mechanical version of [Liouville's theorem,](Liouville's%20theorem%20(mechanics).md) however the [density matrix](density%20matrix.md) is not a [phase space distribution as seen in Liouville's theorem.](Liouville's%20theorem%20(mechanics).md#^8b1efe) %%Tho density matrices can be transformed into quantum phases. iis there a true quantum liouville counterpart in this context? What is the deeper reason they correspond like this???%% ^d4c370 We again see the [correspondence with Liouville's theorem](von%20neumann%20equation.md#^d4c370) since it's useful to rewrite the [von Neumann Equation](von%20Neumann%20Equation.md) in terms of a [Lindbladian superoperator](Lindbladian%20superoperator) such that $\frac{\partial\hat{\rho}(t)}{\partial t} = \hat{\mathcal{L}} \hat{\rho}.$For the untouched von-Neumann equation, $\hat{\mathcal{L}}=-\frac{i}{\hbar}[\hat{H},\hat{\rho}(t)],$ which takes a form reminiscent of a [Liouville operator.](Liouville's%20theorem%20(mechanics).md#^8b1efe) This formulation allows us to then [generalize the von Neumann equation](von%20Neumann%20Equation.md#Generalization%20of%20the%20von%20Neumann%20equation) with other Linbladian superoperators. # Derivation of the von Neumann equation The [von Neumann Equation](von%20Neumann%20Equation.md) can be derived from the expression for a [density matrix that evolves in time under the Schrödinger picture](time%20evolution%20operators.md#Density%20matrix%20transformations) as follows. Consider that we may write any time dependent density operator as [$\hat{\rho}(t)= e^{-i(\hat{H}/\hbar)t}\hat{\rho} e^{i(\hat{H}/\hbar)t}.$](time%20evolution%20operators.md#^9fddeb) Taking its time derivative, we obtain $\frac{d\hat{\rho}(t)}{dt}=\frac{1}{\hbar}(-i\hat{H} e^{-i(\hat{H}/\hbar) t}\hat{\rho}e^{i(\hat{H}/\hbar) t}+i e^{-i(\hat{H}/\hbar) t}\hat{\rho}\hat{H} e^{i(\hat{H}/\hbar) t}) = \frac{i}{\hbar}(\hat{\rho}\hat{H}-\hat{H}\hat{\rho})$$=-\frac{i}{\hbar}[\hat{H},\hat{\rho}(t)]$ %%This Schrödinger picture form needs to be derived elsewhere to make this complete.%% # Solutions of the von Neumann Equation # Generalization of the von Neumann equation #QuanttumMechanics/QuantumDynamics #QuantumMechanics/QuantumStateRepresentations/DensityMatrices