Effective Evolution Equations in Quantum Physics

TL;DR

This paper reviews recent mathematical results on deriving effective evolution equations like the Hartree and Gross-Pitaevskii equations from many-body quantum mechanics, highlighting methods involving BBGKY hierarchy and Fock-space analysis.

Contribution

It provides a comprehensive overview of how effective equations emerge from quantum many-body dynamics, connecting different analytical approaches.

Findings

Derivation of the Hartree equation in the mean field regime.

Approximation of Bose-Einstein condensate dynamics by the Gross-Pitaevskii equation.

Analysis of BBGKY hierarchy and Fock-space methods for deriving effective equations.

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.