Physics:Quantum Liouville equation: Difference between revisions

From HandWiki Test
WikiHarold (talk | contribs)
Bots, Administrators
3,144 edits
imported>WikiHarold
Repair Quantum Collection B backlink template
 
Restore missing Quantum reference definitions
 
(11 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Short description|Quantum Collection topic on Quantum Liouville equation}}
{{Quantum book backlink|Statistical mechanics and kinetic theory}}
{{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>
{{Quantum article nav|previous=Physics:Quantum Distribution functions|previous label=Distribution functions|next=Physics:Quantum Kinetic theory|next label=Kinetic theory}}
<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">


For a closed system with Hamiltonian <math>H</math>, the density operator <math>\rho</math> evolves according to
<div style="width:280px;">
__TOC__
</div>


<math>
<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
i\hbar \frac{\partial \rho}{\partial t} = [H,\rho],
'''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.
</math>
</div>
 
<div style="width:300px;">
[[File:Density matrix.jpg|thumb|280px|Quantum Liouville equation.]]
</div>
 
</div>


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==
==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,
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,
Line 44: Line 52:
</math>
</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.
into the quantum Liouville equation reproduces the Schrödinger equation for <math>|\psi\rangle</math>.<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> Thus, the Liouville equation is a more general framework encompassing both pure and mixed states.


==Formal solution==
==Formal solution==
Line 91: Line 99:


==Open quantum systems==
==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"/>
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">{{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>


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


==References==
= References =
{{reflist|3}}
{{reflist|3}}
{{Author|Harold Foppele}}
{{Author|Harold Foppele}}
Line 102: Line 110:
[[Category:Statistical mechanics]]
[[Category:Statistical mechanics]]


{{Sourceattribution|Quantum Liouville equation|1}}
{{Sourceattribution|Physics:Quantum Liouville equation|1}}

Latest revision as of 00:31, 24 May 2026

← Back to Statistical mechanics and kinetic theory Book I
← Previous : Distribution functions
Next : Kinetic theory →

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. ↑ Shankar, R.. Principles of Quantum Mechanics (2nd ed.). Springer. ISBN 9781475705768. https://link.springer.com/book/10.1007/978-1-4757-0576-8. 
  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. ↑ Breuer, H.-P.; Petruccione, F. (2002). The Theory of Open Quantum Systems. Oxford University Press. ISBN 9780198520634. https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780198520634. 
Author: Harold Foppele

Source attribution: Physics:Quantum Liouville equation

Retrieved from "https://handwiki.scholarlywiki.org/index.php?title=Physics:Quantum_Liouville_equation&oldid=10136"