Limit of classical chaos in quantum systems

The nonlocal effect of quantum mechanics upon the classical chaos around the separatrix in a Hamiltonian system is investigated by extending the definition of the Melnikov function in the semiclassical approximation. It is shown that the quantum correction of the Melnikov function is related to the quantum fluctuation of the energy on the stable and unstable manifolds. This correction is a constant shift of the center of oscillation of the classical Melnikov function from zero. Because of this shift, the effect of quantum mechanics suppresses the classical chaos around the separatrix. Physical estimates are made of the magnitude of the quantum effect for a double-well oscillator system. As examples, we treat the case of the electron for the molecular scale and the proton for the nuclear scale, and also comment on the ammonia molecules ${\mathrm{NH}}_3$, ${\mathrm{ND}}_3$, and ${\mathrm{NT}}_3$.