Effective Evolution Equations from Quantum Dynamics

Abstract

In these notes we review the material presented at the summer school on "Mathematical Physics, Analysis and Stochastics" held at the University of Heidelberg in July 2014. We consider 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.for an appropriate constant C > 0. This gives a mathematically rigorous derivation of the Thomas-Fermi theory, and it tells us how big N should be in order for E T F (N ) to be a good approximation of the true quantum energy (later, better bounds have been obtained [41,58,59]: one knows that the error with respect to Thomas-Fermi theory is of the order N 2 in the limit of large N ).2 Mean-field regime for bosonic systems

Ground states of large bosonic systems : the Gross–Pitaevskii limit revisited

“…It is also the most mathematically demanding regime considered in the literature so far, see [21,17,18] for the derivation of equilibrium states and [10,9,1,24] for dynamics (more extensive lists of references may be found in [20,25,2]). The main reason for this sophistication is the fact that interparticle correlations due to two-body scattering play a leading order role in this regime.…”

Mean-field quantum dynamics for a mixture of Bose–Einstein condensates

“…Here, the map Tr N −1 : B 1 (H N ,sym ) → B 1 (h) is the partial trace from trace class operators on H N to trace class operators on h, defined by Thus, γ (1) N is obtained by "tracing out" N − 1 degrees of freedom from γ N : for example, for a system of N spinless (s = 0) bosons in the pure state Ψ N ,sym ∈ L 2 (R N d , dx 1 …”

“…N ,sim ≡ the wave-functions that are symmetric under permutation of any two variables), the corresponding one-body marginal γ (1) N has kernel Being a density matrix, the one-body marginal γ (1) N has a complete set of real nonnegative eigenvalues that sum up to 1, and being it the partial trace of a many-body state γ N , it is natural to think of these eigenvalues as the occupation numbers in γ N , that is, each eigenvalue of γ (1) N can be interpreted as the fraction of the N particles that are in the same one-body state given by eigenvector associated with the considered eigenvalue.…”

“…In a sense to be specified in the given context, one therefore says that the many-body state γ N exhibits condensation in the state ϕ ∈ h( ϕ = 1) if ϕ is an eigenvector of γ (1) N that belongs to a non-degenerate eigenvalue that is by far larger than all other eigenvalues, i.e., it is almost 1 while all other eigenvalues are almost zero. In other words, γ (1) N ≈ |ϕ ϕ|, the rank-one projection onto ϕ.…”

“…In other words, γ (1) N ≈ |ϕ ϕ|, the rank-one projection onto ϕ. This notion of condensation becomes conceptually well-posed and mathematically rigorous in the limit N → ∞ -a genuine thermodynamic limit, or some simpler prescription on N → ∞ that mimics the thermodynamic limit.…”