Quantum mechanics of Hyperion

This paper is motivated by the suggestion [ W. Zurek Phys. Scri. T 76 186 (1998)] that the chaotic tumbling of the satellite Hyperion would become nonclassical 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 ℏ^2∕3 for chaotic states, and as ℏ² for nonchaotic 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 to ensure the classical limit. Under decoherence, the QC differences in the probability distributions scale as (ℏ²∕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.