Effective Evolution Equations in Quantum Physics

Benjamin Schlein[1]

  • [1] Institute for Applied Mathematics, University of Bonn Endenicher Allee 60, 53115 Bonn

Journées Équations aux dérivées partielles (2011)

  • page 1-19
  • ISSN: 0752-0360

Abstract

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In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.

How to cite

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Schlein, Benjamin. "Effective Evolution Equations in Quantum Physics." Journées Équations aux dérivées partielles (2011): 1-19. <http://eudml.org/doc/219830>.

@article{Schlein2011,
abstract = {In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.},
affiliation = {Institute for Applied Mathematics, University of Bonn Endenicher Allee 60, 53115 Bonn},
author = {Schlein, Benjamin},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
pages = {1-19},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Effective Evolution Equations in Quantum Physics},
url = {http://eudml.org/doc/219830},
year = {2011},
}

TY - JOUR
AU - Schlein, Benjamin
TI - Effective Evolution Equations in Quantum Physics
JO - Journées Équations aux dérivées partielles
DA - 2011/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 19
AB - In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.
LA - eng
UR - http://eudml.org/doc/219830
ER -

References

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