Quantum mechanical tunneling at high energy

Abstract

We study tunneling processes for a quantum mechanical particle in a double-well potential. We show that the instanton estimate of the transition rate between excited states in the two wells is reliable at energies much below the barrier height and weak coupling, in that it is asymptotic to the WKB result. We illustrate how the growth of the amplitude with energy can in a generic basis be perceived as partly due to the growth of available phase space, and we comment on the infrared divergences that appear in the calculation of radiative corrections. We then calculate the rate of tunneling from the ground state in one well to an excited state in the other, induced by an external periodic force, and show that it grows with energy like the geometric mean of the vacuum-to-vacuum and excited-to-excited-state rates. This slower growth is due to the difficulty of converting the energy, initially stored in a single quantum, to large occupation number. We argue by extrapolation that even at energies near the barrier height, induced tunneling should be suppressed by a fraction of the instanton action.