dorsal/arxiv
View SchemaThree Dimensional Confinement : WKB Revisited
| Authors | Anjana Sinha |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0205122 |
| URL | https://arxiv.org/abs/quant-ph/0205122 |
Abstract
An alternate formalism is developed to determine the energy eigenvalues of quantum mechanical systems, confined within a rigid impenetrable spherical box of radius $r_0$, in the framework of Wentzel-Kramers-Brillouin (WKB) approximation. Instead of considering the Langer correction for the centrifugal term, the approach adopted here is that of Hainz and Grabert : The centrifugal term is expanded perturbatively (in powers of $\hbar$), decomposing it into 2 terms -- the classical centrifugal potential and a quantum correction. Hainz and Grabert found that this method reproduced the exact energies of the hydrogen atom, to the first order in $\hbar$, with all higher order corrections vanishing. In the present study, this formalism is extended to the case of radial potentials under hard wall confinement, to check whether the same argument holds good for such confined systems as well. As explicit examples, 3 widely known potentials are studied, which are of considerable importance in the theoretical treatment of various atomic phenomena involving atomic transitions, viz., the 3-dimensional harmonic oscillator, the hydrogen atom, and the Hulthen potential.
{
"annotation_id": "adba001f-f3c7-4a68-809d-fab9aedabf81",
"date_created": "2026-03-02T18:01:51.776000Z",
"date_modified": "2026-03-02T18:01:51.776000Z",
"file_hash": "d8da6cfedb8f8e51f4d367ce11b19be3c8484168941f8b4486352c0b46dac385",
"private": false,
"record": {
"abstract": "An alternate formalism is developed to determine the energy eigenvalues of\nquantum mechanical systems, confined within a rigid impenetrable spherical box\nof radius $r_0$, in the framework of Wentzel-Kramers-Brillouin (WKB)\napproximation. Instead of considering the Langer correction for the centrifugal\nterm, the approach adopted here is that of Hainz and Grabert : The centrifugal\nterm is expanded perturbatively (in powers of $\\hbar$), decomposing it into 2\nterms -- the classical centrifugal potential and a quantum correction. Hainz\nand Grabert found that this method reproduced the exact energies of the\nhydrogen atom, to the first order in $\\hbar$, with all higher order corrections\nvanishing. In the present study, this formalism is extended to the case of\nradial potentials under hard wall confinement, to check whether the same\nargument holds good for such confined systems as well. As explicit examples, 3\nwidely known potentials are studied, which are of considerable importance in\nthe theoretical treatment of various atomic phenomena involving atomic\ntransitions, viz., the 3-dimensional harmonic oscillator, the hydrogen atom,\nand the Hulthen potential.",
"arxiv_id": "quant-ph/0205122",
"authors": [
"Anjana Sinha"
],
"categories": [
"quant-ph"
],
"title": "Three Dimensional Confinement : WKB Revisited",
"url": "https://arxiv.org/abs/quant-ph/0205122"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "2f0601dc-fc66-4fdf-a676-26bbc3703f2b",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}