dorsal/arxiv
View SchemaWigner functions, contact interactions, and matching
| Authors | Mark A. Walton |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0609213 |
| URL | https://arxiv.org/abs/quant-ph/0609213 |
| DOI | 10.1016/j.aop.2006.11.015 |
| Journal | Ann. Phys.322:2233-2248, 2007 |
Abstract
Quantum mechanics in phase space (or deformation quantization) appears to fail as an autonomous quantum method when infinite potential walls are present. The stationary physical Wigner functions do not satisfy the normal eigen equations, the *-eigen equations, unless an ad hoc boundary potential is added [Dias-Prata]. Alternatively, they satisfy a different, higher-order, ``*-eigen-* equation'', locally, i.e. away from the walls [Kryukov-Walton]. Here we show that this substitute equation can be written in a very simple form, even in the presence of an additional, arbitrary, but regular potential. The more general applicability of the -eigen- equation is then demonstrated. First, using an idea from [Fairlie-Manogue], we extend it to a dynamical equation describing time evolution. We then show that also for general contact interactions, the -eigen- equation is satisfied locally. Specifically, we treat the most general possible (Robin) boundary conditions at an infinite wall, general one-dimensional point interactions, and a finite potential jump. Finally, we examine a smooth potential, that has simple but different expressions for x positive and negative. We find that the -eigen- equation is again satisfied locally. It seems, therefore, that the -eigen- equation is generally relevant to the matching of Wigner functions; it can be solved piece-wise and its solutions then matched.
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"abstract": "Quantum mechanics in phase space (or deformation quantization) appears to\nfail as an autonomous quantum method when infinite potential walls are present.\nThe stationary physical Wigner functions do not satisfy the normal eigen\nequations, the *-eigen equations, unless an ad hoc boundary potential is added\n[Dias-Prata]. Alternatively, they satisfy a different, higher-order,\n``*-eigen-* equation\u0027\u0027, locally, i.e. away from the walls [Kryukov-Walton].\nHere we show that this substitute equation can be written in a very simple\nform, even in the presence of an additional, arbitrary, but regular potential.\nThe more general applicability of the -eigen- equation is then demonstrated.\nFirst, using an idea from [Fairlie-Manogue], we extend it to a dynamical\nequation describing time evolution. We then show that also for general contact\ninteractions, the -eigen- equation is satisfied locally. Specifically, we treat\nthe most general possible (Robin) boundary conditions at an infinite wall,\ngeneral one-dimensional point interactions, and a finite potential jump.\nFinally, we examine a smooth potential, that has simple but different\nexpressions for x positive and negative. We find that the -eigen- equation is\nagain satisfied locally. It seems, therefore, that the -eigen- equation is\ngenerally relevant to the matching of Wigner functions; it can be solved\npiece-wise and its solutions then matched.",
"arxiv_id": "quant-ph/0609213",
"authors": [
"Mark A. Walton"
],
"categories": [
"quant-ph"
],
"doi": "10.1016/j.aop.2006.11.015",
"journal_ref": "Ann. Phys.322:2233-2248, 2007",
"title": "Wigner functions, contact interactions, and matching",
"url": "https://arxiv.org/abs/quant-ph/0609213"
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