{
  "annotation_id": "50509581-e66d-4d3e-943c-3149338ad237",
  "date_created": "2026-03-02T18:01:38.558000Z",
  "date_modified": "2026-03-02T18:01:38.558000Z",
  "file_hash": "0cddf8fc2acdfdd94c70e539702e401ccf4eeec26e6abc6099067146d9f56238",
  "private": false,
  "record": {
    "abstract": "An apparent paradox is resolved that concerns the existence of time operators\nwhich have been derived for the quantum harmonic oscillator. There is an\napparent paradox because, although a time operator is canonically conjugate to\nthe Hamiltonian, it has been asserted that no operator exists that is\ncanonically conjugate to the Hamiltonian. In order to resolve the apparent\nparadox, we work in a representation where the phase operator is diagonal. The\nboundary condition on wave functions is such that they be periodic in the phase\nvariable, which is related to the (continuous) eigenvalue of the time operator.\nMatrix elements of the commutator of the time operator with the Hamiltonian\ninvolve the phase variable itself in addition to periodic functions of the\nphase variable. The Hamiltonian is not hermitian when operating in space that\nincludes the phase variable itself. The apparent paradox is resolved when this\nnon-hermeticity is taken into account correctly in the evaluation of matrix\nelements of the commutation relation.",
    "arxiv_id": "quant-ph/0006063",
    "authors": [
      "Alex Granik",
      "H. Ralph Lewis"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "Time operator for the quantum harmonic oscillator: resolution of an   apparent paradox",
    "url": "https://arxiv.org/abs/quant-ph/0006063"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "1d63df6a-0c3f-4ee8-9bc0-32ced9d9e332",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}