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{{Quantum book backlink|Statistical mechanics and kinetic theory}}
'''Quantum Liouville equation''' describes the time evolution of a quantum system in terms of the [[Superoperator|density operator]] rather than a wavefunction.<ref name="vonNeumann">{{cite book |last=von Neumann |first=John |title=Mathematical Foundations of Quantum Mechanics |publisher=Princeton University Press |isbn=9780691178561}}</ref> It is the natural extension of the [[Schrödinger equation]] to statistical ensembles<ref name="LandauLifshitzQM">{{cite book |last1=Landau |first1=L. D. |last2=Lifshitz |first2=E. M. |title=Quantum Mechanics: Non-Relativistic Theory |publisher=Pergamon Press |isbn=9780080209401 |url=https://www.elsevier.com/books/quantum-mechanics/landau/978-0-08-020940-1}}</ref> and is fundamental in [[Physics:quantum statistical mechanics|quantum statistical mechanics]] and the theory of [[Physics:Quantum Open systems|open quantum systems]].<ref name="BreuerPetruccione">{{cite book |last1=Breuer |first1=H.-P. |last2=Petruccione |first2=F. |title=The Theory of Open Quantum Systems |publisher=Oxford University Press |year=2002 |isbn=9780198520634 |url=https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780198520634}}</ref>

For a closed system with Hamiltonian <math>H</math>, the density operator <math>\rho</math> evolves according to

<math>
i\hbar \frac{\partial \rho}{\partial t} = [H,\rho],
</math>

where <math>[H,\rho] = H\rho - \rho H</math> is the commutator.<ref name="Shankar">{{cite book |last=Shankar |first=R. |title=Principles of Quantum Mechanics |edition=2nd |publisher=Springer |isbn=9781475705768 |url=https://link.springer.com/book/10.1007/978-1-4757-0576-8}}</ref><ref name="Dirac">{{cite book |last=Dirac |first=P. A. M. |title=The Principles of Quantum Mechanics |publisher=Oxford University Press |isbn=9780198520115 |url=https://global.oup.com/academic/product/the-principles-of-quantum-mechanics-9780198520115}}</ref> This equation is also known as the '''von Neumann equation'''.<ref name="vonNeumann"/>
[[File:Density matrix.jpg|400px|thumb|Time evolution of the density operator <math>\rho</math> governed by the quantum Liouville equation.]]
==Density operator formalism==
The density operator provides a general description of quantum states.<ref name="Blum">{{cite book |last=Blum |first=K. |title=Density Matrix Theory and Applications |edition=3rd |publisher=Springer |isbn=9783642205606 |url=https://link.springer.com/book/10.1007/978-3-642-20561-3}}</ref> For a pure state,

<math>
\rho = |\psi\rangle \langle \psi|.
</math>

For a mixed state representing an ensemble,

<math>
\rho = \sum_n p_n |\psi_n\rangle \langle \psi_n|,
</math>

with probabilities satisfying

<math>
p_n \ge 0, \quad \sum_n p_n = 1.
</math>

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

<math>
\mathrm{Tr}(\rho) = 1.
</math>

These properties ensure that expectation values of observables can be written as <math>\langle A \rangle = \mathrm{Tr}(\rho A)</math>.<ref name="Blum"/>

==Relation to the Schrödinger equation==
If the system is in a pure state, substituting

<math>
\rho = |\psi\rangle \langle \psi|
</math>

into the quantum Liouville equation reproduces the Schrödinger equation for <math>|\psi\rangle</math>.<ref name="Shankar"/> 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:

<math>
\rho(t) = U(t)\rho(0)U^\dagger(t),
</math>

where

<math>
U(t) = e^{-iHt/\hbar}.
</math>

This evolution preserves trace, Hermiticity, and positivity of the density operator.<ref name="Ballentine">{{cite book |last=Ballentine |first=L. E. |title=Quantum Mechanics: A Modern Development |publisher=World Scientific |year=1998 |isbn=9789810241056 |url=https://worldscientific.com/worldscibooks/10.1142/9645}}</ref>

==Matrix representation==
In the energy eigenbasis, where

<math>
H|n\rangle = E_n |n\rangle,
</math>

the matrix elements evolve as

<math>
i\hbar \frac{d\rho_{mn}}{dt} = (E_m - E_n)\rho_{mn}.
</math>

Hence,

<math>
\rho_{mn}(t) = \rho_{mn}(0)e^{-i(E_m - E_n)t/\hbar}.
</math>

Diagonal elements <math>\rho_{nn}</math> represent populations, while off-diagonal elements <math>\rho_{mn}</math> describe quantum coherences.<ref name="Ballentine"/>

==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 <math>f(q,p,t)</math>.<ref name="Goldstein">{{cite book |last=Goldstein |first=Herbert |title=Classical Mechanics |edition=3rd |publisher=Addison-Wesley |isbn=9780201657029 |url=https://www.pearson.com/en-us/subject-catalog/p/classical-mechanics/P200000003136}}</ref> 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.<ref name="LandauLifshitzStat">{{cite book |last1=Landau |first1=L. D. |last2=Lifshitz |first2=E. M. |title=Statistical Physics |publisher=Pergamon Press |isbn=9780750633727 |url=https://www.elsevier.com/books/statistical-physics/landau/978-0-7506-3372-7}}</ref>

==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.<ref name="Lindblad">{{cite journal |last=Lindblad |first=G. |title=On the generators of quantum dynamical semigroups |journal=Communications in Mathematical Physics |volume=48 |issue=2 |pages=119–130 |year=1976 |doi=10.1007/BF01608499}}</ref><ref name="BreuerPetruccione"/>

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}
{{Author|Harold Foppele}}
[[Category:Quantum mechanics]]
[[Category:Statistical mechanics]]

{{Sourceattribution|Quantum Liouville equation|1}}

Physics:Quantum Liouville equation - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template