Liouville–von Neumann equation
E645944
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Liouville equation | 1 |
| Liouville–von Neumann equation canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7145249 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Liouville–von Neumann equation Context triple: [Bloch equations, relatedTo, Liouville–von Neumann equation]
-
A.
Schrödinger equation
The Schrödinger equation is the fundamental equation of non-relativistic quantum mechanics that governs how the quantum state of a physical system evolves over time.
-
B.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
-
C.
Chapman–Kolmogorov equation
The Chapman–Kolmogorov equation is a fundamental relation in the theory of stochastic processes that expresses how transition probabilities of a Markov process over longer time intervals can be obtained by integrating over intermediate states.
-
D.
Tomonaga–Schwinger equation
The Tomonaga–Schwinger equation is a relativistic generalization of the Schrödinger equation that formulates quantum field evolution on arbitrary spacelike hypersurfaces, forming a key part of covariant quantum field theory.
-
E.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Liouville–von Neumann equation Target entity description: 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.
-
A.
Schrödinger equation
The Schrödinger equation is the fundamental equation of non-relativistic quantum mechanics that governs how the quantum state of a physical system evolves over time.
-
B.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
-
C.
Chapman–Kolmogorov equation
The Chapman–Kolmogorov equation is a fundamental relation in the theory of stochastic processes that expresses how transition probabilities of a Markov process over longer time intervals can be obtained by integrating over intermediate states.
-
D.
Tomonaga–Schwinger equation
The Tomonaga–Schwinger equation is a relativistic generalization of the Schrödinger equation that formulates quantum field evolution on arbitrary spacelike hypersurfaces, forming a key part of covariant quantum field theory.
-
E.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Liouville–von Neumann equation Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
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