dorsal/arxiv
View SchemaSemiclassical expansion of quantum characteristics for many-body potential scattering problem
| Authors | M. I. Krivoruchenko, C. Fuchs, Amand Faessler |
|---|---|
| Categories | |
| ArXiv ID | nucl-th/0605015 |
| URL | https://arxiv.org/abs/nucl-th/0605015 |
| DOI | 10.1002/andp.200610251 |
| Journal | Annalen Phys.16:587-614,2007 |
Abstract
In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star-product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in $\hbar$. We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum-mechanical extensions of the Liouville theorem on the conservation of phase-space volume and the Poincar\'e theorem on the conservation of $2p$ forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior.
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"abstract": "In quantum mechanics, systems can be described in phase space in terms of the\nWigner function and the star-product operation. Quantum characteristics, which\nappear in the Heisenberg picture as the Weyl\u0027s symbols of operators of\ncanonical coordinates and momenta, can be used to solve the evolution equations\nfor symbols of other operators acting in the Hilbert space. To any fixed order\nin the Planck\u0027s constant, many-body potential scattering problem simplifies to\na statistical-mechanical problem of computing an ensemble of quantum\ncharacteristics and their derivatives with respect to the initial canonical\ncoordinates and momenta. The reduction to a system of ordinary differential\nequations pertains rigorously at any fixed order in $\\hbar$. We present\nsemiclassical expansion of quantum characteristics for many-body scattering\nproblem and provide tools for calculation of average values of time-dependent\nphysical observables and cross sections. The method of quantum characteristics\nadmits the consistent incorporation of specific quantum effects, such as\nnon-locality and coherence in propagation of particles, into the semiclassical\ntransport models. We formulate the principle of stationary action for quantum\nHamilton\u0027s equations and give quantum-mechanical extensions of the Liouville\ntheorem on the conservation of phase-space volume and the Poincar\\\u0027e theorem on\nthe conservation of $2p$ forms. The lowest order quantum corrections to the\nKepler periodic orbits are constructed. These corrections show the resonance\nbehavior.",
"arxiv_id": "nucl-th/0605015",
"authors": [
"M. I. Krivoruchenko",
"C. Fuchs",
"Amand Faessler"
],
"categories": [
"nucl-th",
"hep-th",
"math-ph",
"math.DS",
"math.MP",
"quant-ph"
],
"doi": "10.1002/andp.200610251",
"journal_ref": "Annalen Phys.16:587-614,2007",
"title": "Semiclassical expansion of quantum characteristics for many-body potential scattering problem",
"url": "https://arxiv.org/abs/nucl-th/0605015"
},
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