dorsal/arxiv
View SchemaCosine and Sine Operators Related with Orthogonal Polynomial Sets on the Intervall [-1,1]
| Authors | Thomas Appl, Diethard H. Schiller |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0503147 |
| URL | https://arxiv.org/abs/quant-ph/0503147 |
| DOI | 10.1088/0305-4470/38/29/005 |
Abstract
The quantization of phase is still an open problem. In the approach of Susskind and Glogower so called cosine and sine operators play a fundamental role. Their eigenstates in the Fock representation are related with the Chebyshev polynomials of the second kind. Here we introduce more general cosine and sine operators whose eigenfunctions in the Fock basis are related in a similar way with arbitrary orthogonal polynomial sets on the intervall [-1,1]. To each polynomial set defined in terms of a weight function there corresponds a pair of cosine and sine operators. Depending on the symmetry of the weight function we distinguish generalized or extended operators. Their eigenstates are used to define cosine and sine representations and probability distributions. We consider also the inverse arccosine and arcsine operators and use their eigenstates to define cosine-phase and sine-phase distributions, respectively. Specific, numerical and graphical results are given for the classical orthogonal polynomials and for particular Fock and coherent states.
{
"annotation_id": "2ef77ca9-f58d-4206-9c00-2021475fa366",
"date_created": "2026-03-02T18:02:16.746000Z",
"date_modified": "2026-03-02T18:02:16.746000Z",
"file_hash": "c25f25a04418965ceb1201ec33c7063a1ffaed301e70817515dc5e4fe1d282ac",
"private": false,
"record": {
"abstract": "The quantization of phase is still an open problem. In the approach of\nSusskind and Glogower so called cosine and sine operators play a fundamental\nrole. Their eigenstates in the Fock representation are related with the\nChebyshev polynomials of the second kind. Here we introduce more general cosine\nand sine operators whose eigenfunctions in the Fock basis are related in a\nsimilar way with arbitrary orthogonal polynomial sets on the intervall [-1,1].\nTo each polynomial set defined in terms of a weight function there corresponds\na pair of cosine and sine operators. Depending on the symmetry of the weight\nfunction we distinguish generalized or extended operators. Their eigenstates\nare used to define cosine and sine representations and probability\ndistributions. We consider also the inverse arccosine and arcsine operators and\nuse their eigenstates to define cosine-phase and sine-phase distributions,\nrespectively. Specific, numerical and graphical results are given for the\nclassical orthogonal polynomials and for particular Fock and coherent states.",
"arxiv_id": "quant-ph/0503147",
"authors": [
"Thomas Appl",
"Diethard H. Schiller"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/38/29/005",
"title": "Cosine and Sine Operators Related with Orthogonal Polynomial Sets on the Intervall [-1,1]",
"url": "https://arxiv.org/abs/quant-ph/0503147"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "c8cc88f0-f213-41c2-8946-71e814fda8a7",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}