{
  "id": "1711.08350",
  "version": "v1",
  "published": "2017-11-22T15:48:28.000Z",
  "updated": "2017-11-22T15:48:28.000Z",
  "title": "Empirical Measures and Quantum Mechanics: Application to the Mean-Field Limit",
  "authors": [
    "François Golse",
    "Thierry Paul"
  ],
  "comment": "35 pages, no figure",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of $N$-particle systems. We establish an equation governing the evolution of our quantum analogue of the $N$-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Our main application of this new object to the mean-field limit of the $N$-particle Schr\\\"odinger equation is an $O(1/\\sqrt{N})$ convergence rate in some dual Sobolev norm for the Wigner transform of the single-particle marginal of the $N$-particle density operator, uniform in $\\hbar\\in(0,1]$ (where $\\hbar$ is the Planck constant) provided that $V$ and $(-\\Delta)^{3+d/2}V$ have integrable Fourier transforms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-22T15:48:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82C10",
      "35Q55",
      "81Q05"
    ],
    "keywords": [
      "mean-field limit",
      "quantum mechanics",
      "application",
      "quantum analogue",
      "particle density operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}