The phase-space hydrodynamic model for the quantum standard map

Spina, Alejandro; Skodje, Rex T.

doi:10.1016/0010-4655(91)90256-k

Cited by 4 publications

(5 citation statements)

References 31 publications

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“…Using the procedure of Ref. [46] to define |ψ ( p,q, Φ, Π) for this topology, we require that the coherent state |ψ ( p,q) 1 satisfies q|ψ ( p,q) 1 = q + l(2 + L)|ψ ( p,q) 1 , l ∈ Z .…”

Section: Discussionmentioning

confidence: 99%

“…Given a quantum state |ψ , its Husimi distribution H ψ (p, q) is defined by the projection of |ψ onto a coherent state |ψ (p,q) localized around (p, q): H ψ (p, q) ∝ | ψ (p,q) |ψ | 2 . For a system with Euclidean topology, a coherent state localized at (p, q) is a Gaussian with position-space representation localized around q and momentum-space representation localized around p. The system (5) possesses a cylindrical phase-space topology, as the particle position q is a periodic variable and the momentum p ∈ R. In Appendix II, we construct the coherent state for this topology from the Euclidean coherent state [46].…”

Section: Husimi Distributionsmentioning

confidence: 99%

“…For a system with Euclidean topology, a coherent state localized at ͑p , q͒ is a Gaussian with position-space representation localized around q and momentum-space representation localized around p. The system ͑5͒ possesses a cylindrical phase-space topology, as the particle position q is a periodic variable and the momentum p R. In Appendix B, we construct the coherent state for this topology from the Euclidean coherent state. 48 Coherent states provide excellent quantum analogs of classical particles when visualized as wave packets that minimize the position-momentum uncertainty product. The projection onto these particle-like states can thus be viewed intuitively as a sort of classical smearing.…”

Section: Husimi Distributionsmentioning

confidence: 99%

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Quantization of a free particle interacting linearly with a harmonic oscillator

Mainiero

Porter

2007

Chaos

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We investigate the quantization of a free particle coupled linearly to a harmonic oscillator. This system, whose classical counterpart has clearly separated regular and chaotic regions, provides an ideal framework for studying the quantization of mixed systems. We identify key signatures of the classically chaotic and regular portions in the quantum system by constructing Husimi distributions and investigating avoided level crossings of eigenvalues as functions of the strength and range of the interaction between the system's two components. We show, in particular, that the Husimi structure becomes mixed and delocalized as the classical dynamics becomes more chaotic. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2819060͔ Typical classical Hamiltonians systems are neither fully integrable nor fully chaotic, but instead possess mixed dynamics, with islands of stability situated in a chaotic sea. In this paper, we investigate the quantization of a recently-studied system with mixed dynamics.1 This example consists of a free particle that moves around a ring that is divided into two regions. At the boundaries between these regions, the particle is kicked impulsively by a harmonic oscillator (in a manner that conserves the system's total energy), but the particle and oscillator otherwise evolve freely. Although the system is not generic, its separation into regular and chaotic components also allows more precise investigations (both classically and quantum-mechanically) than is typically possible, making this an ideal example to achieve a better understanding of the quantization of mixed systems. By examining avoided level crossings and Husimi distributions in the quantum system, we investigate the quantum signatures of mixed dynamics, demonstrating that the Husimi structures of nearby states become mixed and delocalized as chaos becomes a more prominent feature in the classical phase space.

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Section: Discussionmentioning

confidence: 99%

Section: Husimi Distributionsmentioning

confidence: 99%

Section: Husimi Distributionsmentioning

confidence: 99%

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Quantization of a free particle interacting linearly with a harmonic oscillator

Mainiero

Porter

2007

Chaos

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“…In Hilbert space, the coherent states to be used as coarse-graining functions are, for the cylindrical phase space of each of the two spatial variables i =1, 2, and apart from unnecessary constant phase terms [17],…”

Section: B Husimi Functionsmentioning

confidence: 99%

“…For many years now, Husimi functions [16] have been widely used when comparing quantum and classical systems, as they allow to project quantum functions in phase space in a way that avoids the interpretation problems connected with Wigner functions. In Hilbert space, the coherent states to be used as coarse-graining functions are, for the cylindrical phase space of each of the two spatial variables i = 1, 2, and apart from unnecessary constant phase terms [17],…”

Section: B Husimi Functionsmentioning

confidence: 99%

Quantum double pendulum: Study of an autonomous classically chaotic quantum system

Perotti

2004

Phys. Rev. E

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A numerical study of the quantum double pendulum is conducted. A suitable quantum scaling is found which allows us to have as the only parameters the ratios of the lengths and masses of the two pendula and a (quantum) gravity parameter containing Planck's constant. Comparison with classical and semiclassical results is used to understand the behavior of the energy curves of the levels, to define regimes in terms of the gravity parameter, and to classify the (resonant) interactions among levels by connecting them to various classical phase space structures (resonance islands).

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Quantum Trajectories in Phase Space

Interdisciplinary Applied Mathematics

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The phase-space hydrodynamic model for the quantum standard map

Cited by 4 publications

References 31 publications

Quantization of a free particle interacting linearly with a harmonic oscillator

Quantization of a free particle interacting linearly with a harmonic oscillator

Quantum double pendulum: Study of an autonomous classically chaotic quantum system

Quantum Trajectories in Phase Space

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