To a very good approximation the voltage states of a superconducting tunnel junction can be described by a model of a quantum particle in a potential well and subject to dissipation. In the low-damping limit the dynamics of the particle obeys a master equation developed by Larkin and Ovchinnikov. The stationary solution of this equation was used by them to explain recent experimental data. We extend their work by developing and obtaining the time-dependent solution of the master equation. Thus we are able to study the dependence of the lifetime of the particle inside the well on its initial probability distribution over the quantized energy levels. Among other features, jumps in the lifetime as a function of the height of the potential are obtained. The magnitudes of these jumps are larger if the particle does not start out from the ground state but has instead an initial distribution over the energy levels of the metastable well. In this paper detailed numerical results for different experimental situations are presented. A theoretical understanding of future experimental work involving nonequilibrium situations would involve an examination of the time-dependent solution developed in this paper.