2-DIMENSIONAL TRAJECTORIES OF TUNNELING IN THE SYMMETRICAL DOUBLE-WELL POTENTIAL

Two-dimensional quasiclassical trajectories q1 (q2) are found for an illustrative model of two strongly shifted paraboloids (eigenfrequencies omega-1, omega-2). It is shown that the motion originates along the caustic corresponding to the lower-frequency vibration (q2 = Q2 at omega-1 < omega-2) and proceeds along the classical upside-down barrier trajectory from the characteristic point (Q1', Q2). Increase of omega-1 or the vibrational quantum number n1 leads to diminishing of Q1'-Q1 so that the trajectory tends to the Miller periodic orbit. At omega-1 << omega-2 Q1' approaches the saddle-point value of q1. The statistically averaged trajectory corresponds to an instanton of period i/k(B)T. In the case of central symmetry of the paraboloids, the action along the trajectory obtained corresponds to the product of the Franck-Condon factors. For axially-symmetric positions of the paraboloids the increase in omega-1/omega-2 results in "corner cutting", i.e. deviation of the trajectory from the minimum-energy path.