{
  "id": "1105.0391",
  "version": "v1",
  "published": "2011-05-02T18:07:22.000Z",
  "updated": "2011-05-02T18:07:22.000Z",
  "title": "From a Particle in a Box to the Uncertainty Relation in a Quantum Dot and to Reflecting Walls for Relativistic Fermions",
  "authors": [
    "M. H. Al-Hashimi",
    "U. -J. Wiese"
  ],
  "comment": "36 pages, 5 figures",
  "journal": "Annals of Physics 327 (2012) 2742-2759",
  "categories": [
    "quant-ph",
    "hep-th",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider a 1-parameter family of self-adjoint extensions of the Hamiltonian for a particle confined to a finite interval with perfectly reflecting boundary conditions. In some cases, one obtains negative energy states which seems to violate the Heisenberg uncertainty relation. We use this as a motivation to derive a generalized uncertainty relation valid for an arbitrarily shaped quantum dot with general perfectly reflecting walls in $d$ dimensions. In addition, a general uncertainty relation for non-Hermitean operators is derived and applied to the non-Hermitean momentum operator in a quantum dot. We also consider minimal uncertainty wave packets in this situation, and we prove that the spectrum depends monotonically on the self-adjoint extension parameter. In addition, we construct the most general boundary conditions for semiconductor heterostructures such as quantum dots, quantum wires, and quantum wells, which are characterized by a 4-parameter family of self-adjoint extensions. Finally, we consider perfectly reflecting boundary conditions for relativistic fermions confined to a finite volume or localized on a domain wall, which are characterized by a 1-parameter family of self-adjoint extensions in the $(1+1)$-d and $(2+1)$-d cases, and by a 4-parameter family in the $(3+1)$-d and $(4+1)$-d cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-05-02T18:07:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uncertainty relation",
      "quantum dot",
      "relativistic fermions",
      "reflecting walls",
      "self-adjoint extension"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1016/j.aop.2011.05.003",
      "journal": "Annals of Physics",
      "year": 2012,
      "month": "Jan",
      "volume": 327,
      "number": 1,
      "pages": 1
    },
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 898696,
      "adsabs": "2012AnPhy.327....1A"
    }
  }
}