Quantum dynamics in a cubic potential in the semi-classical limit: Smearing of the homoclinic bifurcation

We consider the dynamics of a particle inside the metastable well of a cubic potential. In the classical picture the particle can oscillate inside the well when its total energy is less than a critical value Ec at which point a homoclinic bifurcation occurs. The quantum particle is always supposed to tunnel out. We set up a semi classical dynamics (which cannot capture tunneling effects) of the quantum particle which can be analyzed in terms of fixed points and stabilities and thus yields quantitative results. We find that the quantum fluctuations smear out the homoclinic bifurcation and for energies greater than a new critical value, substantially less than Ec, the particle always escapes from the well. This implies that even for an initial wave packet centered at the minimum of the well, there is a critical value of the width for which the particle tunnels out even in this semi-classical limit.