dorsal/arxiv
View SchemaDarboux-integration of id\rho/dt=[H,f(\rho)]
| Authors | N. V. Ustinov, S. B. Leble, M. Czachor, M. Kuna |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0005030 |
| URL | https://arxiv.org/abs/quant-ph/0005030 |
| DOI | 10.1016/S0375-9601(01)00013-5 |
| Journal | Phys. Lett. A 279 (2001) 333 |
Abstract
A Darboux-type method of solving the nonlinear von Neumann equation $i\dot \rho=[H,f(\rho)]$, with functions $f(\rho)$ commuting with $\rho$, is developed. The technique is based on a representation of the nonlinear equation by a compatibility condition for an overdetermined linear system. von Neumann equations with various nonlinearities $f(\rho)$ are found to possess the so-called self-scattering solutions. To illustrate the result we consider the Hamiltonian $H$ of a one-dimensional harmonic oscillator and $f(\rho)=\rho^q-2\rho^{q-1}$ with arbitary real $q$. It is shown that self-scattering solutions possess the same asymptotics for all $q$ and that different nonlinearities may lead to effectively indistinguishable evolutions. The result may have implications for nonextensive statistics and experimental tests of linearity of quantum mechanics.
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"abstract": "A Darboux-type method of solving the nonlinear von Neumann equation $i\\dot\n\\rho=[H,f(\\rho)]$, with functions $f(\\rho)$ commuting with $\\rho$, is\ndeveloped. The technique is based on a representation of the nonlinear equation\nby a compatibility condition for an overdetermined linear system. von Neumann\nequations with various nonlinearities $f(\\rho)$ are found to possess the\nso-called self-scattering solutions. To illustrate the result we consider the\nHamiltonian $H$ of a one-dimensional harmonic oscillator and\n$f(\\rho)=\\rho^q-2\\rho^{q-1}$ with arbitary real $q$. It is shown that\nself-scattering solutions possess the same asymptotics for all $q$ and that\ndifferent nonlinearities may lead to effectively indistinguishable evolutions.\nThe result may have implications for nonextensive statistics and experimental\ntests of linearity of quantum mechanics.",
"arxiv_id": "quant-ph/0005030",
"authors": [
"N. V. Ustinov",
"S. B. Leble",
"M. Czachor",
"M. Kuna"
],
"categories": [
"quant-ph",
"nlin.SI"
],
"doi": "10.1016/S0375-9601(01)00013-5",
"journal_ref": "Phys. Lett. A 279 (2001) 333",
"title": "Darboux-integration of id\\rho/dt=[H,f(\\rho)]",
"url": "https://arxiv.org/abs/quant-ph/0005030"
},
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