{
  "annotation_id": "509ff5c1-cdc1-4826-a444-08fd78c658a8",
  "date_created": "2026-03-02T18:02:30.560000Z",
  "date_modified": "2026-03-02T18:02:30.560000Z",
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  "private": false,
  "record": {
    "abstract": "The semiclassical treatment of the two-dimensional harmonic oscillator\nprovides an instructive example of the relation between classical motion and\nthe quantum mechanical energy spectrum. We extend previous work on the\nanisotropic oscillator with incommensurate frequencies and the isotropic\noscillator to the case with commensurate frequencies for which the Lissajous\ncurves appear as classical periodic orbits. Because of the three different\nscenarios depending on the ratio of its frequencies, the two-dimensional\nharmonic oscillator offers a unique way to explicitly analyze the role of\nsymmetries in classical and quantum mechanics.",
    "arxiv_id": "quant-ph/0611062",
    "authors": [
      "Roland Doll",
      "Gert-Ludwig Ingold"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1119/1.2402157",
    "journal_ref": "Am. J. Phys. 75, 208 (2007)",
    "title": "Lissajous curves and semiclassical theory: The two-dimensional harmonic   oscillator",
    "url": "https://arxiv.org/abs/quant-ph/0611062"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "ab1835c0-79ee-4de7-8c00-4b3da6f6c0b6",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}