Fermi's golden rule and exponential decay in two-level systems

The decay of isolated quantum systems is studied. In particular, a specific model for the radiative decay of an excited atomic state is considered. The atom is taken to have two levels and is described by a current distribution $\overrightarrow{\mathrm{j}}(\overrightarrow{\mathrm{x}};1,2)$. Both the case in which the modes of the electromagnetic field are continuous and the case in which they are discrete are considered. Conditions on $\overrightarrow{\mathrm{j}}(\overrightarrow{\mathrm{x}};1,2)$ are found under which the decay law for the excited state is approximately exponential with a lifetime given by Fermi's golden rule. The current should be well localized, not too strong, and not too singular. The $2p\to 1s$ decay in hydrogen is examined in detail. It is also found that the existence of an approximately exponential decay law does not depend upon the ability to extend the resolvent operator of the system to a second Riemann sheet. The case in which the electromagnetic field is quantized in a cavity illustrates this point.