A quantum mechanical treatment of an asymmetric double-well potential (DWP) interacting with a heat bath is presented for circumstances where the contribution of higher vibrational levels to the relaxation dynamics cannot be excluded from consideration. The deep quantum limit characterized by a discrete energy spectrum near the barrier top is considered. The investigation is motivated by simulations on a computer glass which show that the considered parameter regime is "typical" for DWPs responsible for the relaxation peak of sound absorption in glasses. Relaxation dynamics resembling the spatial- and energy-diffusion-controlled limit of the classical Kramers’ problem, and Arrhenius-type behavior is found under specific conditions.