Tunneling from an oscillating state in quantum mechanics

Abstract

We study the decay of general initial states out of a metastable potential well in quantum mechanics. We provide a closed-form expression for the probability current that tunnels through the barrier in terms of the resonant states into which the initial state can be decomposed. All ingredients are computed analytically to first subleading order in the semiclassical limit. Specializing to a coherently-oscillating state, we derive an equivalent semiclassical approximation to the current via the path integral, which circumvents the decomposition into resonant states. We test the accuracy of the approximation by comparing it to a numerical solution of the Schrödinger equation.

arXiv comment29 pages + 3 appendices. v2: major revisions, including a generalization of the main result to oscillations of arbitrary amplitude in the well, a path integral derivation of the main result, and the discussion of small oscillations where a simple closed-form expression is found