Alternative perspective on quantum tunneling and instantons

Abstract

We present an alternative way to compute and interpret quantum tunneling in a one-dimensional double-well potential. For large transition time we show that the quantum action functional gives an analytical expression for tunneling amplitudes. This has been confirmed by numerical simulations giving relative errors in the order of ${10}^{\ensuremath{-}5}$. In contrast to the classical potential, the quantum potential has a triple well if the classical wells are deep enough. Its minima are located at the positions of extrema of the ground state wave function. The striking feature is that a single trajectory with a double instanton reproduces the tunneling amplitude. This is in contrast to the standard instanton approach, where infinitely many instantons and anti-instantons have to be taken into account. The quantum action functional is valid in the deep quantum regime in contrast to the semiclassical regime where the standard instanton approach holds. We compare both approaches via numerical simulations. While the standard instanton picture describes only the transition between potential minima of equal depth, the quantum action may give rise to instantons also for asymmetric potential minima. Such a case is illustrated by an example.