Relaxation to Quantum Statistical Equilibrium of the Wigner-Weisskopf Atom in a One-Dimensional Radiation Field. II. Finite Systems

In this paper, a model is studied in which it was possible to obtain an expression for the exact solution of the master equation for the problem of spontaneous emission in a finite system. The model chosen for discussion is that of the Wigner-Weisskopf atom in interaction with a massless boson field, here 1-dimensional. The solution takes the form of a constant term plus a time-dependent one, expressed as the sum of residues at a series of poles along the real axis of a Laplace transform variable. Numerical calculations were performed on various aspects of the solution, and although these were too delicate to be quite certain around sensitive values of the time, the general picture is clear: After an initial decay to a value near zero, the series gave rise at fairly regular intervals to rapid and large fluctuations, the size of which never quite attains the initial value, but may nonetheless be large even after very long times. This result seems to be in agreement with the observation made by Zwanzig, namely, that for finite systems the master equation might demonstrate properties associated with the finite size of the system which would become important over certain long-time scales and would be ignored by the use of the customary ``thermodynamic limit'' of statistical mechanics. The relationship between this work and that of Montroll and Mazur and of Rubin is also discussed.