{
  "id": "quant-ph/0612032",
  "version": "v2",
  "published": "2006-12-05T12:43:03.000Z",
  "updated": "2007-07-02T09:24:48.000Z",
  "title": "A New Look at the Quantum Mechanics of the Harmonic Oscillator",
  "authors": [
    "H. A. Kastrup"
  ],
  "comment": "100 pages (Latex); v2: abstract simplified, remarks and references added",
  "journal": "AnnalenPhys.16:439-528,2007",
  "doi": "10.1002/andp.200610245",
  "categories": [
    "quant-ph",
    "astro-ph",
    "hep-th",
    "math-ph",
    "math.MP",
    "physics.atom-ph"
  ],
  "abstract": "Classically the Harmonic Oscillator (HO) is the generic example for the use of angle and action variables phi in R mod 2 pi and I > 0. But the symplectic transformation (\\phi,I) to (q,p) is singular for (q,p) = (0,0). Globally {(q,p)} has the structure of the plane R^2, but {(phi,I)} that of the punctured plane R^2 -(0,0). This implies qualitative differences for the QM of the two phase spaces: The quantizing group for the plane R^2 consists of the (centrally extended) translations generated by {q,p,1}, but the corresponding group for {(phi,I)} is SO(1,2) = Sp(2,R)/Z_2, (Sp(2,R): symplectic group of the plane), with Lie algebra basis {h_0 = I, h_1 = I cos phi, h_2 = - I sin phi}. In the QM for the (phi,I)-model the three h_j correspond to self-adjoint generators K_j, j=0,1,2, of irreducible unitary representations (positive discrete series) for SO(1,2) or one of its infinitely many covering groups, the Bargmann index k > 0 of which determines the ground state energy E (k, n=0) = hbar omega k of the (phi,I)-Hamiltonian H(K). For an m-fold covering the lowest possible value is k=1/m, which can be made arbitrarily small! The operators Q and P, now expressed as functions of the K_j, keep their usual properties, but the richer structure of the K_j quantum model of the HO is ``erased'' when passing to the simpler Q,P model! The (phi,I)-variant of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with the (Landau) levels of charged particles in magnetic fields, with the propagation of light in vacuum, passing through electric or magnetic fields. Finally it leads to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-07-02T09:24:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03.65.Fd",
      "42.50.Xa",
      "03.65.Ge"
    ],
    "keywords": [
      "harmonic oscillator",
      "quantum mechanics",
      "magnetic fields",
      "lie algebra basis",
      "action variables phi"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Annalen der Physik",
      "year": 2007,
      "month": "Jul",
      "volume": 519,
      "number": "7-8",
      "pages": 439
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 100,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 733673,
      "adsabs": "2007AnP...519..439K"
    }
  }
}