A discrete formulation of the theory of sojourn times in a two-state system

Most theoretical analyses of single-molecule spectroscopy (SMS) are formulated in terms of a first-order isomerization process. The mathematical problem to which the analysis reduces requires one to find the probability density for the total residence time in one of the states at a given observation time T. In this paper we develop a slightly more rigorous theory in which account is taken of the discrete pulsing of the laser. It is shown how to pass to the continuum limit. We develop analogous results to analyze experiments in which the quantum yield is less than 1.