dorsal/arxiv
View SchemaQuantum Correction in Exact Quantization Rules
| Authors | Zhong-Qi Ma, Bo-Wei Xu |
|---|---|
| Categories | |
| ArXiv ID | physics/0502109 |
| URL | https://arxiv.org/abs/physics/0502109 |
| DOI | 10.1209/epl/i2004-10418-8 |
| Journal | Europhys. Lett. 2005 March |
Abstract
An exact quantization rule for the Schr\"{o}dinger equation is presented. In the exact quantization rule, in addition to $N\pi$, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second P\"{o}schl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom.
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"abstract": "An exact quantization rule for the Schr\\\"{o}dinger equation is presented. In\nthe exact quantization rule, in addition to $N\\pi$, there is an integral term,\ncalled the quantum correction. For the exactly solvable systems we find that\nthe quantum correction is an invariant, independent of the number of nodes in\nthe wave function. In those systems, the energy levels of all the bound states\ncan be easily calculated from the exact quantization rule and the solution for\nthe ground state, which can be obtained by solving the Riccati equation. With\nthis new method, we re-calculate the energy levels for the one-dimensional\nsystems with a finite square well, with the Morse potential, with the symmetric\nand asymmetric Rosen-Morse potentials, and with the first and the second\nP\\\"{o}schl-Teller potentials, for the harmonic oscillators both in one\ndimension and in three dimensions, and for the hydrogen atom.",
"arxiv_id": "physics/0502109",
"authors": [
"Zhong-Qi Ma",
"Bo-Wei Xu"
],
"categories": [
"physics.comp-ph",
"physics.atom-ph"
],
"doi": "10.1209/epl/i2004-10418-8",
"journal_ref": "Europhys. Lett. 2005 March",
"title": "Quantum Correction in Exact Quantization Rules",
"url": "https://arxiv.org/abs/physics/0502109"
},
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