Calculation of the Lifetime of a Metastable System

We have compared, in the context of a soluble model, two approximate methods of calculating the lifetime of a metastable state with the "exact" method. The exact method involves the usual relation $T=2\mathrm{}\hslash {\Gamma }^{-1\mathrm{}}$ between the lifetime $T$ of a metastable state and the displacement $\frac{\Gamma }{2}$ of the resonance pole of the $S$ matrix from the real energy axis. This result is contrasted with those obtained by using Fermi's "golden rule" and F. T. Smith's time-delay method. While the "golden rule" method involves a first-order determination of the transition amplitude connecting the approximate bound state to the final approximate continuum state, Smith's method is nonperturbative and utilizes the $S$ matrix directly.