Lifetime Matrix in Collision Theory

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, 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 ₈ are normalized to unit total flux in and out through a sphere at R, the matrix elements are Q₈₉=limit of{^r<R₈{₉}^*d-R ({v₈}^-1₈₉+₊S₈₊{v₊}^-1{S₉₊}^*) }₀ₕasR, where the average value is taken to eliminate oscillating terms at large R, S₈₊ is an element of the unitary scattering matrix, S, and v₈ is the velocity in the ith channel. Q is Hermitian; a diagonal element Q₈₈ is the average lifetime of a collision beginning in the ith channel. As a function of the energy Q is related to S: Q=-i d{S^}dE; Q and S contain the same information, from complementary points of view. When Q is diagonalized, its proper values, q₈₈, are the lifetimes of metastable states if they are large compared to E; for a sharp resonance, the measured lifetime is the average of q₈₈ (E) over a distribution in energy. The corresponding eigenfunctions, ₈, are the proper functions to describe these metastable states. The causality principle appears directly from an inequality involving the integral expression for Q₈₈ or q₈₈, and it is shown how some of its consequences for inelastic collisions can be deduced.