Abstract

An analytical solution to the Wigner-Weisskopf problem (an excited two-level atom in inter-action with a radiation field), obtained for both finite-and infinite-length boxes, is re-examined in terms of the qualitative behavior implied. As the equations and the plots of the solutions show, there is a major difference between the behavior of the finite system (with discrete spec-trum) and that obtaining in the large-system limit. In the first case, a pulse-shaped wave "travels down the line" and comes back (and is sent off again) many times, completely "losing its shape" in the process (and subsequently re-gaining it on a much longer time scale infinitely often, due to the presence of a Poincare recurrence). In the large-system limit, on the other hand, a delta-impulse-like wave travels down the line only once (in finite time), and there is also no loss of shape upon its return after infinite time. Thus, there is no longer any even temporary "smearing out" of the initially sharply localized energy, and hence no "mixing" in the intuitive sense of the word. Nonetheless a dense spectrum is found (similarly as in the distribution theoretical case of an isolated delta-impulse in an infinite domain), and hence weak mixing in the sense of Lebowitz. The contradiction can be resolved at the expense of having to abandon some symmetry: by assuming the atom adjacent to two cavities of incommensurate lengths. Then the infinite system limit is unchanged (no return in finite time), but the transition is characterized by intuitive mixing of increasing effectiveness.