On the Mean Field and Classical Limits of Quantum Mechanics

Abstract

Abstract. The main result in this paper is a new inequality bearing on solutions of the N -body linear Schrödinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of N identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of C 1,1 interaction potentials. The quantity measuring the approximation of the N -body quantum dynamics by its mean field limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with exponent 2. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13 (1979), 115-123]. Our approach of this problem is based on a direct analysis of the N -particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization.In memory of Louis Boutet de Monvel (1941Monvel ( -2014 Statement of the problemIn nonrelativistic quantum mechanics, the dynamics of N identical particles of mass m in R d is described by the linear Schrödinger equationwhere the unknown is Ψ ≡ Ψ(t, x 1 , . . . , x N ) ∈ C, the N -particle wave function, while x 1 , x 2 , . . . , x N designate the positions of the 1st, 2nd,. . . , N th particle. The interaction between the kth and lth particles is given by the potential V , a realvalued measurable function defined a.e. on R d , such thatDenoting the macroscopic length scale by L > 0, we define a time scale T > 0 such that the total interaction energy of the typical particle with the N −1 other particles is of the order of m(L/T ) 2 . With the dimensionless space and time variables defined asx := x/L ,t := t/T , the interaction potential is scaled aŝ V (ẑ) := N T 2 mL 2 V (z) .1991 Mathematics Subject Classification. 82C10, 35Q55 (82C05,35Q83).

The Schrödinger Equation in the Mean-Field and Semiclassical Regime

“…This convergence holds independently of distinguished scaling assumptions that would link N and , as a consequence of the main result in [9], and of Theorem IV.2 in [18], which establishes the validity of the classical limit of the Hartree equation, leading to the Vlasov equation (the right vertical arrow in Diagram 1).…”

“…(In other words, the coupling constant is chosen so that the kinetic and potential energies of typical N -particle configurations are of the same order of magnitude.) After suitable rescalings of the various quantities involved (see the introduction of [9] for details), the dynamics happens to be governed by a two-parameter family of Schrödinger equations indexed by and N , of the form …”

“…In [9], the limits corresponding to the horizontal arrows have been studied in detail. Following an idea of Dobrushin [8], we arrived at quantitative estimates on distances metrizing the weak convergence of probability measures, or of their quantum analogues.…”

“…Following an idea of Dobrushin [8], we arrived at quantitative estimates on distances metrizing the weak convergence of probability measures, or of their quantum analogues. However, the method used in [9] differs from Dobrushin's in [8] in several ways. First, we avoid formulating our results in terms of the N -particle phase-space empirical measure, which does not seem to have any clear quantum analogue.…”

“…Our approach of the Vlasov limit of the N -particle Liouville equation is of a completely Eulerian nature, so that its extension to the quantum dynamics is rather straightforward. The main result in [9] is that the upper horizontal arrow, i.e. the large N , mean-field limit of the N -particle linear Schrödinger equation to the Hartree equation is uniform as → 0.…”