Classical Limit and Chaotic Regime in a Semi-Quantum Hamiltonian

Based on a quantum dynamical invariant of motion, I, we study the classical limit of a semiclassical Hamiltonian composed by a full quantum harmonic oscillator plus a classical particle plus a "semiclassical" coupling quartic term. The motion-invariant is closely related to the uncertainty principle. The classical limit (CL) is determined by the relationship between I and the total energy of the system, defining an adimensional invariant E r . We find that the CL coincides with the results of a purely classical treatment. Both invariants allow to follow the transit between quantum nonchaotic to the classical chaotic regime. Particularly, with E r we define the threshold above which chaos appears, and the interval during which both regimes co-exist.