The WKB approach to quantum tunneling from a metastable well is generalized to include a weak oscillatory force. Both one-dimensional decay and multidimensional decay, appropriate for a system coupled to a dissipative environment, are considered. In both cases, general expressions are derived for the enhancement of the decay rate linear in the applied force. The enhancement is expressed exclusively in terms of the instanton trajectories of the unperturbed system. For a cubic potential with no damping, the decay rate exhibits sharp resonances when the applied frequency coincides with the energy spacing between the ground and excited states in the metastable well. In the strong-damping limit, this resonance structure is completely suppressed.
The double-well potential is arguably one of the most important potentials in quantum mechanics, because the solution contains the notion of a state as a linear superposition of 'classical' states, a concept which has become very important in quantum information theory. It is therefore desirable to have solutions to simple double-well potentials that are accessible to the undergraduate student. We describe a method for obtaining the numerically exact eigenenergies and eigenstates for such a model, along with the energies obtained through the Wentzel–Kramers–Brillouin (WKB) approximation. The exact solution is accessible with elementary mathematics, though numerical solutions are required. We also find that the WKB approximation is remarkably accurate, not just for the ground state, but also for the excited states.
The dissipationless quantum-thermal escape rate from a metastable well is often written as a Boltzmann average of the Hill-Wheeler flux over a continuum of energies. This continuum treatment diverges exponentially at low temperature, because it pairs the quantum partition function of the well with a classical flux integral. Retaining the discrete nature of the quasibound spectrum removes the divergence exactly, and the resulting rate is obtained in closed form on both sides of the crossover temperature
T0. Evaluating a uniform Kemble transmission probability on Bohr-Sommerfeld levels further removes the parabolic-barrier error and recovers the WKB decay rate of the lowest resonance at zero temperature. Benchmarked against the resonances of cubic and quintic potentials, obtained by complex scaling, this uniform semiclassical rate stays within
9% of the exact one over eleven orders of magnitude. Below
T0, the rate rises above its zero-temperature plateau with the level spacing as activation energy, and the plateau ends at a temperature
T1 noticeably below
T0. In an isolated well, whose levels are not replenished, the thermal rate is only the initial slope of a decay which ceases to be single exponential at
T1, while at
T0 the initial escape moves from tunneling inside the well to the barrier top. We discuss implications for Josephson junctions, single-electron pumps, and nuclear burning in dense stellar matter.