Time‐dependent density operator in the random impulse approximation: I. Connections with the “generalized sudden approximation” for photoionization and electron impact ionization; and II. Relevance to reduced density operators, particle–hole matrices, and propagators for the initial state

Abstract

AbstractA density operator analysis is made of the particle‐ejection (e.g., photoionization) problem in the “generalized sudden approximation” for the case in which the ejected particle is detected but its dynamical properties are not observed. It is shown that the resultant statistical density operator may be derived from that obtained in a particle‐conserving Hamiltonian‐evolution linear‐response model with a random impulse perturbation, and that the physical applicability of the latter model embraces the range of applicability of the former. The initial value of the statistical density operator for the above particle‐ejection problem is shown to be the N – (1)‐particle reduced density operator of the initial state; and the initial value of the statistical density operator for the excitations in the random impulse model is shown to be the N‐particle Hermitian operator whose matrix representation is the G matrix of Garrod and Percus. The significance of the eigenvalue spectrum of these operators to the excitation properties of the system is discussed, especially for the random impulse model, where a large eigenvalue of the G matrix can signal strong preferential excitation to its corresponding particle‐hole collective state, even for a random perturbation. Extensions of these ideas to excitations from states with a large eigenvalue of the two‐particle reduced density operator (e.g., superconducting states) are mentioned. The The applicability of these density matrices to the description of the excitation spectrum due to a well‐defined perturbation is discussed. The relationship of these time‐dependent density operators to the one‐particle propagator and the particle–hole propagator (polarization propagator) is established.