Liouville-von Neumann Equation

Available identifiers

WikidataQ2533076 ScholiaQ2533076MaRDI QIDQ6534349

describes how a density operator (for pure or for mixed states) evolves in time

Description

Just as the Schrödinger equation describes how pure states evolve in time, the Liouville-von Neumann equation describes how a density operator evolves in time. Note that there can be different density operators for pure or for mixed states.
Note the similarity with the classical Liouville equation where the commutator brackets [.,.] are replaced by Poisson brackets {.,.}

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Defining Formula: $\frac{\partial \hat{\rho }}{\partial t}=-\frac{\mathrm{i}}{\hslash }[\hat{H},\hat{\rho }]$

$\hat{H}$ symbol represents Quantum Hamiltonian Operator

$\hat{\rho }$ symbol represents Quantum Density Operator

$\hslash$ symbol represents Planck Constant

$t$ symbol represents Time

$\mathrm{i}$ symbol represents imaginary unit

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Related URL

https://en.wikipedia.org/wiki/Density_matrix#Von_Neumann_equation_for_time_evolution

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