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Liouville equation quantum Liouville equation describes the time evolution of a quantum system in terms of the density operator rather than a wavefunction. It is the natural extension of the Schrödinger equation to statistical ensembles and is fundamental in quantum statistical mechanics and the theory of open quantum systems. This equation is also known as the von Neumann equation. The density operator provides a general description of quantum states. For a mixed state representing an ensemble, The density operator is Hermitian, positive semi-definite, and normalized: If the system is in a pure state, substituting Thus, the Liouville equation is a more general framework encompassing both pure and mixed states.

Quantum Liouville equation.

Density operator formalism

The density operator provides a general description of quantum states.[1] For a pure state,

ρ=|ψ⟩⟨ψ|.

For a mixed state representing an ensemble,

ρ=∑npn|ψn⟩⟨ψn|,

with probabilities satisfying

pn≥0,∑npn=1.

The density operator is Hermitian, positive semi-definite, and normalized:

Tr(ρ)=1.

These properties ensure that expectation values of observables can be written as ⟨A⟩=Tr(ρA).[1]

Relation to the Schrödinger equation

If the system is in a pure state, substituting

ρ=|ψ⟩⟨ψ|

into the quantum Liouville equation reproduces the Schrödinger equation for |ψ⟩.[2] Thus, the Liouville equation is a more general framework encompassing both pure and mixed states.

Formal solution

For a time-independent Hamiltonian, the solution can be written using the unitary time-evolution operator:

ρ(t)=U(t)ρ(0)U†(t),

where

U(t)=e−iHt/ℏ.

This evolution preserves trace, Hermiticity, and positivity of the density operator.[3]

Matrix representation

In the energy eigenbasis, where

H|n⟩=En|n⟩,

the matrix elements evolve as

iℏdρmndt=(Em−En)ρmn.

Hence,

ρmn(t)=ρmn(0)e−i(Em−En)t/ℏ.

Diagonal elements ρnn represent populations, while off-diagonal elements ρmn describe quantum coherences.[3]

Connection to classical Liouville equation

The quantum Liouville equation is the operator analogue of the classical Liouville equation, which governs the evolution of a phase-space distribution function f(q,p,t).[4] The correspondence is established via:

  • Classical dynamics: Poisson brackets
  • Quantum dynamics: commutators

In the classical limit, commutators reduce to Poisson brackets, providing a bridge between classical and quantum statistical mechanics.[5]

Open quantum systems

For open systems interacting with an environment, the evolution is no longer purely unitary. The quantum Liouville equation is generalized to master equations such as the Lindblad equation, which include dissipative and decoherence effects.[6][7]

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ 1.0 1.1 Blum, K.. Density Matrix Theory and Applications (3rd ed.). Springer. ISBN 9783642205606. https://link.springer.com/book/10.1007/978-3-642-20561-3. 
  2. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Shankar
  3. ↑ 3.0 3.1 Ballentine, L. E. (1998). Quantum Mechanics: A Modern Development. World Scientific. ISBN 9789810241056. https://worldscientific.com/worldscibooks/10.1142/9645. 
  4. ↑ Goldstein, Herbert. Classical Mechanics (3rd ed.). Addison-Wesley. ISBN 9780201657029. https://www.pearson.com/en-us/subject-catalog/p/classical-mechanics/P200000003136. 
  5. ↑ Landau, L. D.; Lifshitz, E. M.. Statistical Physics. Pergamon Press. ISBN 9780750633727. https://www.elsevier.com/books/statistical-physics/landau/978-0-7506-3372-7. 
  6. ↑ Lindblad, G. (1976). "On the generators of quantum dynamical semigroups". Communications in Mathematical Physics 48 (2): 119–130. doi:10.1007/BF01608499. 
  7. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named BreuerPetruccione
Author: Harold Foppele

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