dorsal/arxiv
View SchemaLyapunov exponent in quantum mechanics. A phase-space approach
| Authors | V. I. Man'ko, R. Vilela Mendes |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0002049 |
| URL | https://arxiv.org/abs/quant-ph/0002049 |
| DOI | 10.1016/S0167-2789(00)00117-2 |
| Journal | Physica D145 (2000) 330-348 |
Abstract
Using the symplectic tomography map, both for the probability distributions in classical phase space and for the Wigner functions of its quantum counterpart, we discuss a notion of Lyapunov exponent for quantum dynamics. Because the marginal distributions, obtained by the tomography map, are always well defined probabilities, the correspondence between classical and quantum notions is very clear. Then we also obtain the corresponding expressions in Hilbert space. Some examples are worked out. Classical and quantum exponents are seen to coincide for local and non-local time-dependent quadratic potentials. For non-quadratic potentials classical and quantum exponents are different and some insight is obtained on the taming effect of quantum mechanics on classical chaos. A detailed analysis is made for the standard map. Providing an unambiguous extension of the notion of Lyapunov exponent to quantum mechnics, the method that is developed is also computationally efficient in obtaining analytical results for the Lyapunov exponent, both classical and quantum.
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"abstract": "Using the symplectic tomography map, both for the probability distributions\nin classical phase space and for the Wigner functions of its quantum\ncounterpart, we discuss a notion of Lyapunov exponent for quantum dynamics.\nBecause the marginal distributions, obtained by the tomography map, are always\nwell defined probabilities, the correspondence between classical and quantum\nnotions is very clear. Then we also obtain the corresponding expressions in\nHilbert space. Some examples are worked out. Classical and quantum exponents\nare seen to coincide for local and non-local time-dependent quadratic\npotentials. For non-quadratic potentials classical and quantum exponents are\ndifferent and some insight is obtained on the taming effect of quantum\nmechanics on classical chaos. A detailed analysis is made for the standard map.\nProviding an unambiguous extension of the notion of Lyapunov exponent to\nquantum mechnics, the method that is developed is also computationally\nefficient in obtaining analytical results for the Lyapunov exponent, both\nclassical and quantum.",
"arxiv_id": "quant-ph/0002049",
"authors": [
"V. I. Man\u0027ko",
"R. Vilela Mendes"
],
"categories": [
"quant-ph",
"astro-ph",
"nlin.CD"
],
"doi": "10.1016/S0167-2789(00)00117-2",
"journal_ref": "Physica D145 (2000) 330-348",
"title": "Lyapunov exponent in quantum mechanics. A phase-space approach",
"url": "https://arxiv.org/abs/quant-ph/0002049"
},
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