{
  "annotation_id": "3e0b542a-6df7-4932-ac52-6a321027b745",
  "date_created": "2026-03-02T18:02:41.222000Z",
  "date_modified": "2026-03-02T18:02:41.222000Z",
  "file_hash": "b6817f32729805116c373697d9dbc7ebd314f3b621f016c06023437038417b80",
  "private": false,
  "record": {
    "abstract": "Requirements of a conjugate operator are emphasized, especially in its role\nin uncertainty relations.It is argued that in many contexts it is necessary to\nextend the Hilbert space in order to define a conjugate operator as in gauge\ntheories. Example of a particle in a box is analysed. This is closely related\nto the quantum oscillator through cosine states of Susskind and Glogower.It is\nused to justify London\u0027s phase wave functions albeit as part of a larger\nHilbert space. A new definition phase uncertainty neccessiated by periodicity\nis proposed.It is close to the usual r.m.s. definition.Corresponding number-\nphase uncertainty relation is obtained and its implications are discussed.\nHilbert space of an oscillator is identified with the Hilbert space of a planar\nrotor with a $Z_2$ gauge invariance.This is used to construct states analogous\nto the cosine and sine states and to illustrate unitary equivalence of Hilbert\nspaces.",
    "arxiv_id": "quant-ph/9710020",
    "authors": [
      "H. S. Sharatchandra"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "Phase of the quantum oscillator",
    "url": "https://arxiv.org/abs/quant-ph/9710020"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "a1098cb3-f4f3-4990-8ff1-da56c9a27f26",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}