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.
@incollection{JEDP_2011____A11_0,
author = {Schlein, Benjamin},
title = {Effective {Evolution} {Equations} in {Quantum} {Physics}},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {11},
pages = {1--19},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2011},
doi = {10.5802/jedp.83},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jedp.83/}
}
TY - JOUR AU - Schlein, Benjamin TI - Effective Evolution Equations in Quantum Physics JO - Journées équations aux dérivées partielles PY - 2011 SP - 1 EP - 19 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.83/ DO - 10.5802/jedp.83 LA - en ID - JEDP_2011____A11_0 ER -
Schlein, Benjamin. Effective Evolution Equations in Quantum Physics. Journées équations aux dérivées partielles (2011), article no. 11, 19 p. doi : 10.5802/jedp.83. http://www.numdam.org/articles/10.5802/jedp.83/
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