Tunneling phenomena in three-dimensional double-well potentials

The tunneling of a hydrogen atom through the barrier of a three‐dimensional double‐well potential is considered. From the time‐dependent Schrödinger equation, expressions are derived for the ensemble‐averaged probability density and for the probability that the hydrogen atom is in the reactant region, in the barrier region, or in the product region. It is found that when thermal vibrations are not taken into account, the ensemble‐averaged probability density may oscillate with time about its equilibrium value. When thermal vibrations are included, the oscillations become damped and the probability density approaches equilibrium. The tunneling rate is found to decrease considerably for increasing barrier thickness and barrier height.