The three particle collision operator in quantum mechanics

Abstract

We analyse the quantum mechanical collision operator for three incident free particles from the point of view of transport theory. Starting from the Liouville-von Neumann equation for the density matrix, in the form introduced previously by Prigogine and coworkers, we obtain the generalization to quantum systems of the well known Choh-Uhlenbeck result for classical genuine triple collisions. This expression, written in terms of the Heisenberg operators of motion exp[— iHt] for two and three particles, is free of the divergence difficulties occurring in standard collision theory when the limits of a large system and of long times are taken in order to define a three particle transition probability. We also discuss the connection of this result with the Lippman-Schwinger theory and show that the scattering operator for three incident particles is not expressible as the square of a scattering matrix, like in the two particle problem.