Effective Evolution Equations from Quantum Dynamics

  • Niels Benedikter
  • Marcello Porta
  • Benjamin Schlein

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 7)

Table of contents

  1. Front Matter
    Pages i-vii
  2. Niels Benedikter, Marcello Porta, Benjamin Schlein
    Pages 1-6
  3. Niels Benedikter, Marcello Porta, Benjamin Schlein
    Pages 7-16
  4. Niels Benedikter, Marcello Porta, Benjamin Schlein
    Pages 17-29
  5. Niels Benedikter, Marcello Porta, Benjamin Schlein
    Pages 31-36
  6. Niels Benedikter, Marcello Porta, Benjamin Schlein
    Pages 37-56
  7. Niels Benedikter, Marcello Porta, Benjamin Schlein
    Pages 57-78
  8. Niels Benedikter, Marcello Porta, Benjamin Schlein
    Pages 79-86
  9. Back Matter
    Pages 87-91

About this book


These notes investigate the time evolution of quantum systems, and in particular the rigorous derivation of effective equations approximating the many-body Schrödinger dynamics in certain physically interesting regimes. The focus is primarily on the derivation of time-dependent effective theories (non-equilibrium question) approximating many-body quantum dynamics. The book is divided into seven sections, the first of which briefly reviews the main properties of many-body quantum systems and their time evolution. Section 2 introduces the mean-field regime for bosonic systems and explains how the many-body dynamics can be approximated in this limit using the Hartree equation. Section 3 presents a method, based on the use of coherent states, for rigorously proving the convergence towards the Hartree dynamics, while the fluctuations around the Hartree equation are considered in Section 4. Section 5 focuses on a discussion of a more subtle regime, in which the many-body evolution can be approximated by means of the nonlinear Gross-Pitaevskii equation. Section 6 addresses fermionic systems (characterized by antisymmetric wave functions); here, the fermionic mean-field regime is naturally linked with a semiclassical regime, and it is proven that the evolution of approximate Slater determinants can be approximated using the nonlinear Hartree-Fock equation. In closing, Section 7 reexamines the same fermionic mean-field regime, but with a focus on mixed quasi-free initial data approximating thermal states at positive temperature.           


Bosonic Mean-field Regime Coherent States Approach Fermionic Mean-field Regime Gross-Pitaevskii Equation Hartree Equation Hartree-Fock Equation Many-body Quantum Dynamics Many-body Quantum Systems Many-body Schrödinger Dynamics Schrödinger Dynamics Slater Determinants

Authors and affiliations

  • Niels Benedikter
    • 1
  • Marcello Porta
    • 2
  • Benjamin Schlein
    • 3
  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  2. 2.Institute of MathematicsUniversity of ZürichZürichSwitzerland
  3. 3.Institute of MathematicsUniversity of ZürichZürichSwitzerland

Bibliographic information

  • DOI
  • Copyright Information The Author(s) 2016
  • Publisher Name Springer, Cham
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-3-319-24896-7
  • Online ISBN 978-3-319-24898-1
  • Series Print ISSN 2197-1757
  • Series Online ISSN 2197-1765
  • Buy this book on publisher's site