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Lyapunov exponent in quantum mechanics. A phase-space approach

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V. I. Man'ko, R. Vilela Mendes (Grupo de Fisica Matematica, Univ. Lisboa)

Quantum Physics · Submitted 17 Feb 2000

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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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arXiv IDquant-ph/0002049 (version 1)
Pith IDY6TQKJY6, Pith's permanent record of this paper and its review
CitationR. Vilela Mendes (Grupo de Fisica Matematica, Univ. Lisboa), V. I. Man'ko. "Lyapunov exponent in quantum mechanics. A phase-space approach." arXiv:quant-ph/0002049 (2000). https://pith.science/paper/Y6TQKJY6
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,144 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.28 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.19 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. contribution 0.81 0.89 0.13 asserted 0.91
s1 Because the marginal distributions, obtained by the tomography map, are always well defined probabilities, the correspondence between classical and quantum notions is very clear. interpretation 0.80 0.51 0.07 asserted 1.00
s2 Then we also obtain the corresponding expressions in Hilbert space. contribution 0.60 0.89 0.04 asserted 0.98
s3 Some examples are worked out. contribution 0.63 0.90 0.02 asserted 0.81
s4 Classical and quantum exponents are seen to coincide for local and non-local time-dependent quadratic potentials. result 1.00 0.89 0.23 asserted 1.00
s5 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. result 0.74 0.89 0.28 asserted 0.97
s6 A detailed analysis is made for the standard map. contribution 0.84 0.90 0.05 asserted 0.89
s7 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. contribution 0.91 0.91 0.49 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.88 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original 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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