Quantum-to-classical limit in a Hamiltonian system

The dynamics of quantum and classical probability distributions are computed for a model of two coupled rotors (or pendulums), with emphasis on the ℏ scaling of the quantum-classical (QC) differences. This scaling is not the same for coarse-grained and fine-grained quantities. The QC differences in the averages of observables scale as ℏ^2∕3 for chaotic states, but as ℏ² for regular states. The QC differences in probability distributions scale as ℏ^1∕3 for chaotic states. No simple scaling is found for those differences in regular states, although their overall magnitudes are similar to those in chaotic states. QC differences arise first in the short-wavelength regime, and subsequently spread to all wavelengths. A satisfactory classical limit is obtained without invoking environmental decoherence, which in any case would be effective in suppressing only short-wavelength QC differences.