Quantum mechanics of classically non-integrable systems

Eckhardt, Bruno

doi:10.1016/0370-1573(88)90130-5

Cited by 353 publications

(107 citation statements)

References 329 publications

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“…[3][4][5][6] Are the signatures of quantum chaos quantum manifestations of classical nonintegrability or manifestations of quantum nonintegrability? The second alternative implies the existence of a quantum integrability condition which can stand on its own, i.e.…”

Section: Introductionmentioning

confidence: 99%

Quantum integrability and action operators in spin dynamics

Weigert¹,

Müller²

1995

Chaos, Solitons & Fractals

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A new formulation of the quantum integrability condition for spin systems is proposed. It eliminates the ambiguities inherent in formulations derived from a direct transcription of the classical integrability criterion. In the new formulation, quantum integrability of an N -spin system depends on the existence of a unitary transformation which expresses the Hamiltonian as a function of N action operators. All operators are understood to be algebraic expressions of the spin-components with no restriction to any finite-dimensional matrix representation. The consequences of quantum (non-)integrability on the structure of quantum invariants are discussed in comparison with the consequences of classical (non-)integrability on the corresponding classical invariants. Our results indicate that quantum integrability is universal for systems with N = 1 and contingent for systems with N ≥ 2.

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Section: Introductionmentioning

confidence: 99%

Quantum integrability and action operators in spin dynamics

Weigert¹,

Müller²

1995

Chaos, Solitons & Fractals

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“…In this case, the breakdown of Berry's formula is a direct consequence of the absolute degeneracy of the phase appearing in the time-integral of Eq. (3), that is a4>jat = E-H(zA) = E -· H(z). Not only do the endpoints of the contributing midpoint trajectories lie on the periodic orbit itself but they are also required to be on the same straight segment as z.…”

Section: • -3-mentioning

confidence: 99%

“…2 On the other hand, the eigenfunc~ tions of strongly chaotic systems were believed for a long time to correspond to Wigner functions which are homogeneous over the energy shell. 3 In billiards, this conjecture leads to eigenfunctions which in turn are homogeneous over configuration space. This simple picture.…”

mentioning

confidence: 99%

Scars in billiards: The phase space approach

et al. 1990

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“…In recent years, many quantum nonintegrability effects have been identified and studied [8][9][10]. The most prominent among them is perhaps the striking correlation between the fluctuation properties of the energy spectrum of a quantum model system and the (non-)integrability of the corresponding classical model [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

“…The most prominent among them is perhaps the striking correlation between the fluctuation properties of the energy spectrum of a quantum model system and the (non-)integrability of the corresponding classical model [10][11][12]. However, there is at present no positive evidence for deterministic randomness in any property of nonintegrable quantum systems.…”

Section: Introductionmentioning

confidence: 99%

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

Srivastava

Müller

1990

Z. Physik B - Condensed Matter

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We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.

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Quantum mechanics of classically non-integrable systems

Cited by 353 publications

References 329 publications

Quantum integrability and action operators in spin dynamics

Quantum integrability and action operators in spin dynamics

Scars in billiards: The phase space approach

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

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