arXiv subjects

L. E. Ballentine

Publications and source records attributed to L. E. Ballentine.

3 recordsLinked to original sources

arXiv2005

The Quantum Mechanics of Hyperion

This paper is motivated by the suggestion [W. Zurek, Physica Scripta, T76, 186 (1998)] that the chaotic tumbling of the satellite Hyperion would become non-classical within 20 years, but for the effects of environmental decoherence. The dynamics of quantum and classical probability distributions are compared for a satellite rotating perpendicular to its orbital plane, driven by the gravitational gradient. The model is studied with and without environmental decoherence. Without decoherence, the maximum quantum-classical (QC) differences in its average angular momentum scale as hbar^{2/3} for chaotic states, and as hbar^2 for non-chaotic states, leading to negligible QC differences for a macroscopic object like Hyperion. The quantum probability distributions do not approach their classical limit smoothly, having an extremely fine oscillatory structure superimposed on the smooth classical background. For a macroscopic object, this oscillatory structure is too fine to be resolved by any realistic measurement. Either a small amount of smoothing (due to the finite resolution of the apparatus) or a very small amount of environmental decoherence is sufficient ensure the classical limit. Under decoherence, the QC differences in the probability distributions scale as (hbar^2/D)^{1/6}, where D is the momentum diffusion parameter. We conclude that decoherence is not essential to explain the classical behavior of macroscopic bodies.

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arXiv2001

Characteristics of Quantum-Classical Correspondence for Two Interacting Spins

The conditions of quantum-classical correspondence for a system of two interacting spins are investigated. Differences between quantum expectation values and classical Liouville averages are examined for both regular and chaotic dynamics well beyond the short-time regime of narrow states. We find that quantum-classical differences initially grow exponentially with a characteristic exponent consistently larger than the largest Lyapunov exponent. We provide numerical evidence that the time of the break between the quantum and classical predictions scales as log(${\cal J}/ \hbar$), where ${\cal J}$ is a characteristic system action. However, this log break-time rule applies only while the quantum-classical deviations are smaller than order hbar. We find that the quantum observables remain well approximated by classical Liouville averages over long times even for the chaotic motions of a few degree-of-freedom system. To obtain this correspondence it is not necessary to introduce the decoherence effects of a many degree-of-freedom environment.

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arXiv2001

Quantum-Classical Correspondence for the Equilibrium Distributions of Two Interacting Spins

We consider the quantum and classical Liouville dynamics of a non-integrable model of two coupled spins. Initially localised quantum states spread exponentially to the system dimension when the classical dynamics are chaotic. The long-time behaviour of the quantum probability distributions and, in particular, the parameter-dependent rates of relaxation to the equilibrium state are surprisingly well approximated by the classical Liouville mechanics even for small quantum numbers. As the accessible classical phase space becomes predominantly chaotic, the classical and quantum probability equilibrium configurations approach the microcanonical distribution, although the quantum equilibrium distributions exhibit characteristic `minimum' fluctuations away from the microcanonical state. The magnitudes of the quantum-classical differences arising from the equilibrium quantum fluctuations are studied for both pure and mixed (dynamically entangled) quantum states. In both cases the standard deviation of these fluctuations decreases as $(\hbar/{\mathcal J})^{1/2}$, where ${\mathcal J}$ is a measure of the system size. In conclusion, under a variety of conditions the differences between quantum and classical Liouville mechanics are shown to become vanishingly small in the classical limit (${\mathcal J}/\hbar \to \infty$) of a non-dissipative model endowed with only a few degrees of freedom.

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