Tunneling in double-well potentials within Nelson's stochastic mechanics: Application to ammonia inversion
Nelson's stochastic mechanics formulates quantum dynamics as a real-time conservative diffusion process in which a particle undergoes Brownian-like motion with a fluctuation amplitude fixed by Planck's constant. While being mathematically equivalent to the Schrödinger formulation, this approach provides an alternative dynamical framework that enables the study of time-resolved quantities that are not straightforwardly defined within the standard operator-based approach. In the present work, Nelson's stochastic mechanics is employed to investigate tunneling-time statistics for bound states in double-well potentials. Using first-passage time theory within this framework, both the mean tunneling time, $\barτ$, and the full probability distribution, $p(τ)$, are computed. The theoretical predictions are validated through extensive numerical simulations of stochastic trajectories for two representative potentials. For the square double-well potential, analytical expressions for $\barτ$ are derived and are shown to be in excellent agreement with simulations. In the high-barrier limit, the results reveal a direct relation between the stochastic-mechanical and quantum-mechanical tunneling times, expressed as $τ_{\mathrm{QM}} = (π/2)\barτ$, where $τ_{\mathrm{QM}}$ corresponds to half the oscillation period of the probability of finding the particle in either well. This relation is further confirmed for generic double-well systems through a WKB analysis. As a concrete application, the inversion dynamics of the ammonia molecule is analyzed, yielding an inversion frequency of approximately 24 GHz, in close agreement with experimental observations. These results highlight the potential of stochastic mechanics as a conceptually coherent and quantitatively consistent framework for analyzing tunneling phenomena in quantum systems.