Lifetime Matrix in Collision Theory

Abstract

The duration of a collision is usually a rather ill-defined concept, depending on a more or less arbitrary choice of a collision distance. If the collision lifetime is defined as the limit, as $R\ensuremath{\rightarrow}\ensuremath{\infty}$, of the difference between the time the particles spend within a distance $R$ of each other and the time they would have spent there in the absence of the interaction, a well-defined quantity emerges which is finite as long as the interaction vanishes rapidly enough at large $R$.In quantum mechanics, using steady-state wave functions, the average time of residence in a region is the integrated density divided by the total flux in (or out), and the lifetime is defined as the difference between these residence times with and without interaction. Transformation properties require construction of the lifetime matrix, Q. If the wave functions ${\ensuremath{\psi}}_{i}$ are normalized to unit total flux in and out through a sphere at $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the matrix elements are ${Q}_{\mathrm{ij}}=limit of\text{}{\left[\ensuremath{\int}{}^{r