Relaxation to Quantum-Statistical Equilibrium of the Wigner-Weisskopf Atom in a One-Dimensional Radiation Field. III. The Quantum-Mechanical Solution

Abstract

A study is undertaken to cast light on difficulties, which arose in the first two papers of this series, pertaining to the occurrence of negative probabilities in the weak‐coupling solution of the generalized Prigogine‐Résibois master equation for the model of the Wigner‐Weisskopf atom in a one‐dimensional radiation field. The Schrödinger equation is solved exactly for the model with the initial condition for spontaneous emission, and then the weak‐coupling approximations to the solution, both for an infinite and for a finite system, are derived as inverse Laplace transform integrals. An extensive analysis, theoretical and numerical, of these is undertaken, and comparison is made with the corresponding results based on the master equation. In particular, quantitative estimates of the Poincaré recurrence times for finite systems are made. It is found that considerable differences exist between the statistical‐mechanical and quantum‐mechanical results, but that both manifest nonanalyticity in the coupling parameter as it tends to zero. Suggestions are given for further work toward the resolution of these discrepancies and a better understanding of the weak‐coupling limit.