Physics:Quantum Liouville equation - Revision history

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	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==
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	==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}}

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	'''~~Quantum~~ Liouville equation''' describes the time evolution of a quantum system in terms of the 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 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>~~		'''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.

	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"/>~~
	</div>		</div>

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	'''Quantum Liouville equation''' describes the time evolution of a quantum system in terms of the 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 Liouville equation''' describes the time evolution of a quantum system in terms of the 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 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		For a closed system with Hamiltonian <math>H</math>, the density operator <math>\rho</math> evolves according to

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	'''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 Liouville equation''' describes the time evolution of a quantum system in terms of the 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		For a closed system with Hamiltonian <math>H</math>, the density operator <math>\rho</math> evolves according to

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