TEMPORAL BEHAVIOR OF QUANTUM-MECHANICAL SYSTEMS

The temporal behavior of quantum mechanical systems is reviewed. We ma inly focus our attention on the time development of the so-called ''su rvival'' probability of those systems that are initially prepared in e igenstates of the unperturbed Hamiltonian, by assuming that the latter has a continuous spectrum. The exponential decay of the survival prob ability, familiar, for example, in radioactive decay phenomena, is rep resentative of a purely probabilistic character of the system under co nsideration and is naturally expected to lead to a master equation. Th is behavior, however, can be found only at intermediate times, for dev iations from it exist both at short and long times and can have signif icant consequences. After a short introduction to the long history of the research on the temporal behavior of such quantum mechanical syste ms, the short-time behavior and its controversial consequences when it is combined with von Neumann's projection postulate in quantum measur ement theory are critically overviewed from a dynamical point of view. We also discuss the so-called quantum Zeno effect from this standpoin t. The behavior of the survival amplitude is then scrutinized by inves tigating the analytic properties of its Fourier and Laplace transforms . The analytic property that there is no singularity except a branch c ut running along the real energy axis in the first Riemannian sheet is an important reflection of the time-reversal invariance of the dynami cs governing the whole process. It is shown that the exponential behav ior is due to the presence of a simple pole in the second Riemannian s heet, while the contribution of the branch point yields a power behavi or for the amplitude. The exponential decay form is cancelled at short times and dominated at very long times by the branch-point contributi ons, which give a Gaussian behavior for the former and a power behavio r for the latter. In order to realize the exponential law in quantum t heory, it is essential to take into account a certain kind of macrosco pic nature of the total system, since the exponential behavior is rega rded as a manifestation of a complete loss of coherence of the quantum subsystem under consideration. In this respect, a few attempts at ext racting the exponential decay form on the basis of quantum theory, aim ing at the master equation, are briefly reviewed, including van Hove's pioneering work and his well-known ''lambda(2)T'' limit. In the attem pt to further clarify the mechanism of the appearance of a purely prob abilistic behavior without resort to any approximation, a solvable dyn amical model is presented and extensively studied. The model describes an ultrarelativistic particle interacting with N two-level systems (c alled ''spins'') and is shown to exhibit an exponential behavior at al l times in the weak-coupling, macroscopic limit. Furthermore, it is sh own that the model can even reproduce the short-time Gaussian behavior followed by the exponential law when an appropriate initial state is chosen. The analysis is exact and no approximation is involved. An int erpretation for the change of the temporal behavior in quantum systems is drawn from the results obtained. Some implications for the quantum measurement problem are also discussed, in particular in connection w ith dissipation.