dorsal/arxiv
View SchemaPath Integral Approach for Spaces of Non-constant Curvature in Three Dimensions
| Authors | Christian Grosche |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0511135 |
| URL | https://arxiv.org/abs/quant-ph/0511135 |
| DOI | 10.1134/S1063778807030131 |
| Journal | Phys.Atom.Nucl.70:537-544,2007 |
Abstract
In this contribution I show that it is possible to construct three-dimensional spaces of non-constant curvature, i.e. three-dimensional Darboux-spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that in the two three-dimensional Darboux spaces, which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In $\threedDI$ we find seven coordinate systems which separate the Schr\"odinger equation. For the second space, $\threedDII$, all coordinate systems of flat three-dimensional Euclidean space which separate the Schr\"odinger equation also separate the Schr\"odinger equation in $\threedDII$. I solve the path integral on $\threedDI$ in the $(u,v,w)$-system, and on $\threedDII$ in the $(u,v,w)$-system and in spherical coordinates.
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"abstract": "In this contribution I show that it is possible to construct\nthree-dimensional spaces of non-constant curvature, i.e. three-dimensional\nDarboux-spaces. Two-dimensional Darboux spaces have been introduced by Kalnins\net al., with a path integral approach by the present author. In comparison to\ntwo dimensions, in three dimensions it is necessary to add a curvature term in\nthe Lagrangian in order that the quantum motion can be properly defined. Once\nthis is done, it turns out that in the two three-dimensional Darboux spaces,\nwhich are discussed in this paper, the quantum motion is similar to the\ntwo-dimensional case. In $\\threedDI$ we find seven coordinate systems which\nseparate the Schr\\\"odinger equation. For the second space, $\\threedDII$, all\ncoordinate systems of flat three-dimensional Euclidean space which separate the\nSchr\\\"odinger equation also separate the Schr\\\"odinger equation in\n$\\threedDII$. I solve the path integral on $\\threedDI$ in the $(u,v,w)$-system,\nand on $\\threedDII$ in the $(u,v,w)$-system and in spherical coordinates.",
"arxiv_id": "quant-ph/0511135",
"authors": [
"Christian Grosche"
],
"categories": [
"quant-ph"
],
"doi": "10.1134/S1063778807030131",
"journal_ref": "Phys.Atom.Nucl.70:537-544,2007",
"title": "Path Integral Approach for Spaces of Non-constant Curvature in Three Dimensions",
"url": "https://arxiv.org/abs/quant-ph/0511135"
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