A quantum analog to the classical quasiperiodic motion

Abstract

A quantum mechanical mnemonic which identifies the energy levels corresponding to those obtained by quantization of the classical quasiperiodic motion in coupled anaharmonic oscillator systems is presented. The mnemonic reproduces the known classical regular to irregular behavior as a function of the energy. The mnemonic is based on the definition of a suitable effective Hamiltonian for each level. Quasiperiodic levels are those in which the wave function has a weight greater than 50% on a degenerate subspace of any separable part of the Hamiltonian. This condition ensures that there exists an effective Hamiltonian defined on this degenerate subspace which commutes with the separable Hamiltonian; the latter is therefore an effective constant of motion for the total Hamiltonian. In some cases, depending on the degenerate subspace, other effective constants of motion exist. There is a correlation between the semiclassical quantization methods of quasiperiodic trajectories and the present effective Hamiltonian approach.