dorsal/arxiv
View SchemaSpin-1/2 square-root operator equation with Coulomb potential - a perturbation analysis
| Authors | Tobias Gleim |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0601211 |
| URL | https://arxiv.org/abs/quant-ph/0601211 |
Abstract
Seeking for a relativistic generalisation of the non-relativistic Schroedinger equation, one very soon arrives at equations with a square-root operator by having applied the quantum mechanical correspondence principle to the formula of relativistic energy. The problems of these equations are at least two fold: when coupled to an electromagnetic field, their relativistic invariance is not evident or even doubtful and due to their non-local character, it seems to be that they cannot be maintained mathematically in an easy way. For spin-1/2 particles, these difficulties can be overcome by the Dirac equation, which leads e.g. to the prediction of binding energies of an electron in a hydrogen atom that are compatible with experimental results up to the forth order of the fine structure constant, inclusively. Ignoring the problem with relativistic invariance, one may ask, if there exists a square-root equation, for which one can achieve the same good agreement with experiments for the latter physical system. It is going to be shown, that the answer to this question is affirmative and does not exceed the skills obtained in a course about non-relativistic quantum mechanics and physics of atoms, respectively.
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"abstract": "Seeking for a relativistic generalisation of the non-relativistic\nSchroedinger equation, one very soon arrives at equations with a square-root\noperator by having applied the quantum mechanical correspondence principle to\nthe formula of relativistic energy. The problems of these equations are at\nleast two fold: when coupled to an electromagnetic field, their relativistic\ninvariance is not evident or even doubtful and due to their non-local\ncharacter, it seems to be that they cannot be maintained mathematically in an\neasy way. For spin-1/2 particles, these difficulties can be overcome by the\nDirac equation, which leads e.g. to the prediction of binding energies of an\nelectron in a hydrogen atom that are compatible with experimental results up to\nthe forth order of the fine structure constant, inclusively. Ignoring the\nproblem with relativistic invariance, one may ask, if there exists a\nsquare-root equation, for which one can achieve the same good agreement with\nexperiments for the latter physical system. It is going to be shown, that the\nanswer to this question is affirmative and does not exceed the skills obtained\nin a course about non-relativistic quantum mechanics and physics of atoms,\nrespectively.",
"arxiv_id": "quant-ph/0601211",
"authors": [
"Tobias Gleim"
],
"categories": [
"quant-ph"
],
"title": "Spin-1/2 square-root operator equation with Coulomb potential - a perturbation analysis",
"url": "https://arxiv.org/abs/quant-ph/0601211"
},
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