dorsal/arxiv
View SchemaOn infinite walls in deformation quantization
| Authors | S. Kryukov, M. A. Walton |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0412007 |
| URL | https://arxiv.org/abs/quant-ph/0412007 |
| DOI | 10.1016/j.aop.2004.12.004 |
| Journal | Annals Phys.317:474-491,2005 |
Abstract
We examine the deformation quantization of a single particle moving in one dimension (i) in the presence of an infinite potential wall, (ii) confined by an infinite square well, and (iii) bound by a delta function potential energy. In deformation quantization, considered as an autonomous formulation of quantum mechanics, the Wigner function of stationary states must be found by solving the so-called $\*$-genvalue (``stargenvalue'') equation for the Hamiltonian. For the cases considered here, this pseudo-differential equation is difficult to solve directly, without an ad hoc modification of the potential. Here we treat the infinite wall as the limit of a solvable exponential potential. Before the limit is taken, the corresponding $\*$-genvalue equation involves the Wigner function at momenta translated by imaginary amounts. We show that it can be converted to a partial differential equation, however, with a well-defined limit. We demonstrate that the Wigner functions calculated from the standard Schr\"odinger wave functions satisfy the resulting new equation. Finally, we show how our results may be adapted to allow for the presence of another, non-singular part in the potential.
{
"annotation_id": "aeaaa180-ba55-4b74-9851-71a36bda2669",
"date_created": "2026-03-02T18:02:13.318000Z",
"date_modified": "2026-03-02T18:02:13.318000Z",
"file_hash": "7dc878922025d75878ebe31dc02f2c3493b753156c0fbd01917f50ceaece4ecf",
"private": false,
"record": {
"abstract": "We examine the deformation quantization of a single particle moving in one\ndimension (i) in the presence of an infinite potential wall, (ii) confined by\nan infinite square well, and (iii) bound by a delta function potential energy.\nIn deformation quantization, considered as an autonomous formulation of quantum\nmechanics, the Wigner function of stationary states must be found by solving\nthe so-called $\\*$-genvalue (``stargenvalue\u0027\u0027) equation for the Hamiltonian.\nFor the cases considered here, this pseudo-differential equation is difficult\nto solve directly, without an ad hoc modification of the potential. Here we\ntreat the infinite wall as the limit of a solvable exponential potential.\nBefore the limit is taken, the corresponding $\\*$-genvalue equation involves\nthe Wigner function at momenta translated by imaginary amounts. We show that it\ncan be converted to a partial differential equation, however, with a\nwell-defined limit. We demonstrate that the Wigner functions calculated from\nthe standard Schr\\\"odinger wave functions satisfy the resulting new equation.\nFinally, we show how our results may be adapted to allow for the presence of\nanother, non-singular part in the potential.",
"arxiv_id": "quant-ph/0412007",
"authors": [
"S. Kryukov",
"M. A. Walton"
],
"categories": [
"quant-ph"
],
"doi": "10.1016/j.aop.2004.12.004",
"journal_ref": "Annals Phys.317:474-491,2005",
"title": "On infinite walls in deformation quantization",
"url": "https://arxiv.org/abs/quant-ph/0412007"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "a91fce82-a086-4fe7-9d73-9fdf228c3a7d",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}