dorsal/arxiv
View SchemaUltraviolet analysis of one dimensional quantum systems
| Authors | Marco Frasca |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0202067 |
| URL | https://arxiv.org/abs/quant-ph/0202067 |
| Journal | Nuovo Cim.B117:867-874,2002 |
Abstract
Starting from the study of one-dimensional potentials in quantum mechanics having a small distance behavior described by a harmonic oscillator, we extend this way of analysis to models where such a behavior is not generally expected. In order to obtain significant results we approach the problem by a renormalization group method that can give a fixed point Hamiltonian that has the shape of a harmonic oscillator. In this way, good approximations are obtained for the ground state both for the eigenfunction and the eigenvalue for problems like the quartic oscillator, the one-dimensional Coulomb potential having a not normalizable ground state solution and for the one-dimensional Kramers-Henneberger potential. We keep a coupling constant in the potential and take it running with a generic cut-off that goes to infinity. The solution of the Callan-Symanzik equation for the coupling constant generates the harmonic oscillator Hamiltonian describing the behavior of the model at very small distances (ultraviolet behavior). This approach, although algorithmic in its very nature, does not appear to have a simple extension to obtain excited state behavior. Rather, it appears as a straightforward non-perturbative method.
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"abstract": "Starting from the study of one-dimensional potentials in quantum mechanics\nhaving a small distance behavior described by a harmonic oscillator, we extend\nthis way of analysis to models where such a behavior is not generally expected.\nIn order to obtain significant results we approach the problem by a\nrenormalization group method that can give a fixed point Hamiltonian that has\nthe shape of a harmonic oscillator. In this way, good approximations are\nobtained for the ground state both for the eigenfunction and the eigenvalue for\nproblems like the quartic oscillator, the one-dimensional Coulomb potential\nhaving a not normalizable ground state solution and for the one-dimensional\nKramers-Henneberger potential. We keep a coupling constant in the potential and\ntake it running with a generic cut-off that goes to infinity. The solution of\nthe Callan-Symanzik equation for the coupling constant generates the harmonic\noscillator Hamiltonian describing the behavior of the model at very small\ndistances (ultraviolet behavior). This approach, although algorithmic in its\nvery nature, does not appear to have a simple extension to obtain excited state\nbehavior. Rather, it appears as a straightforward non-perturbative method.",
"arxiv_id": "quant-ph/0202067",
"authors": [
"Marco Frasca"
],
"categories": [
"quant-ph",
"hep-th",
"math-ph",
"math.AP",
"math.MP",
"physics.atom-ph"
],
"journal_ref": "Nuovo Cim.B117:867-874,2002",
"title": "Ultraviolet analysis of one dimensional quantum systems",
"url": "https://arxiv.org/abs/quant-ph/0202067"
},
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