{
  "annotation_id": "491f7d5e-6f48-45a7-a53a-e9e890d77e77",
  "date_created": "2026-03-02T18:01:42.418000Z",
  "date_modified": "2026-03-02T18:01:42.418000Z",
  "file_hash": "fd5f5121854736ca0ef58b3f5ad83111aacc3d047806bcf474371a6aa5ba2c1f",
  "private": false,
  "record": {
    "abstract": "The proof of the Heisenberg uncertainty relation is modified to produce two\nimprovements: (a) the resulting inequality is stronger because it includes the\ncovariance between the two observables, and (b) the proof lifts certain\nrestrictions on the state to which the relation is applied, increasing its\ngenerality. The restrictions necessary for the standard inequality to apply are\nnot widely known, and they are discussed in detail. The classical analog of the\nHeisenberg relation is also derived, and the two are compared. Finally, the\nmodified relation is used to address the apparent paradox that eigenfunctions\nof the z component of angular momentum L_z do not satisfy the \\phi-L_z\nHeisenberg relation; the resolution is that the restrictions mentioned above\nmake the usual inequality inapplicable to these states. The modified relation\ndoes apply, however, and it is shown to be consistent with explicit\ncalculations.",
    "arxiv_id": "quant-ph/0011115",
    "authors": [
      "Eric D. Chisolm"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1119/1.1317561",
    "journal_ref": "Am J Phys 69, 368-371 (2001)",
    "title": "Generalizing the Heisenberg uncertainty relation",
    "url": "https://arxiv.org/abs/quant-ph/0011115"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "863ef0f0-8154-4ca9-b921-e233448b78cb",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}