# On the Mean Field and Classical Limits of Quantum Mechanics

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 equations. 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.
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Submitted on : Friday, November 13, 2015 - 12:19:39 PM
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François Golse, Clément Mouhot, Thierry Paul. On the Mean Field and Classical Limits of Quantum Mechanics. Communications in Mathematical Physics, Springer Verlag, 2016, 343, pp.165-205. ⟨10.1007/s00220-015-2485-7⟩. ⟨hal-01119132v4⟩

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