Liouville–von Neumann equation

E645944

equation of motion evolution equation quantum mechanical equation

The Liouville–von Neumann equation is the fundamental quantum-mechanical evolution equation governing the time dependence of the density operator, generalizing the Schrödinger equation to mixed states and open-system dynamics.

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Observed surface forms (1)

Surface form	Occurrences
Liouville equation	1

Statements (48)

Predicate	Object
instanceOf	equation of motion ⓘ evolution equation ⓘ quantum mechanical equation ⓘ
alsoKnownAs	quantum Liouville equation NERFINISHED ⓘ von Neumann equation NERFINISHED ⓘ
appliesTo	closed quantum systems ⓘ mixed quantum states ⓘ open quantum systems ⓘ pure quantum states ⓘ
assumes	Hamiltonian generates time evolution ⓘ
category	fundamental equation of quantum theory ⓘ
classicalAnalogue	classical Liouville equation NERFINISHED ⓘ
codomain	space of density operators on a Hilbert space ⓘ
contains	commutator of Hamiltonian and density operator ⓘ
describes	conservation of probability in quantum mechanics ⓘ reversible quantum dynamics for closed systems ⓘ
domain	Hilbert space of the quantum system ⓘ
ensures	unitary time evolution for closed systems ⓘ
expresses	unitary time evolution in terms of the density operator ⓘ
field	quantum mechanics ⓘ
framework	density matrix formalism ⓘ
generalizes	Schrödinger equation NERFINISHED ⓘ
governs	time dependence of the density matrix ⓘ time evolution of the density operator ⓘ
historicallyNamedAfter	John von Neumann NERFINISHED ⓘ Joseph Liouville NERFINISHED ⓘ
implies	Hermiticity of density operator is preserved ⓘ positivity of density operator is preserved for closed systems ⓘ trace of density operator is conserved ⓘ
isBasisFor	Lindblad master equation in Markovian open systems NERFINISHED ⓘ
isFirstOrderIn	time ⓘ
isLinearIn	density operator ⓘ
mathematicalForm	operator differential equation ⓘ
reducesTo	Schrödinger equation for pure states in state-vector form NERFINISHED ⓘ
relatedTo	Heisenberg equation of motion NERFINISHED ⓘ master equations for open quantum systems ⓘ
requires	self-adjoint Hamiltonian operator ⓘ
standardForm	iħ dρ/dt = [H, ρ] ⓘ
timeDependentVersionOf	quantum Liouville equation NERFINISHED ⓘ
usedIn	nuclear magnetic resonance ⓘ open quantum systems theory ⓘ quantum information theory ⓘ quantum optics ⓘ quantum statistical mechanics ⓘ quantum thermodynamics ⓘ
usesSymbol	H for the Hamiltonian operator ⓘ ħ for the reduced Planck constant ⓘ ρ for the density operator ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Bloch equations → relatedTo → Liouville–von Neumann equation ⓘ

Vlasov equation (for long-range interactions and negligible collisions) → relatedTo → Liouville–von Neumann equation ⓘ

subject surface form: Vlasov equation

this entity surface form: Liouville equation