Temporal behavior of quantum mechanical systems

Abstract

The temporal behavior of quantum mechanical systems is reviewed. We study the so-called quantum Zeno effect, that arises from the quadratic short-time behavior, and the analytic properties of the ``survival" amplitude. It is shown that the exponential behavior is due to the presence of a simple pole in the second Riemannian sheet, while the contribution of the branch point yields a power behavior for the amplitude. The exponential decay form is cancelled at short times and dominated at very long times by the branch-point contributions, which give a Gaussian behavior for the former and a power behavior for the latter. In order to realize the exponential law in quantum theory, it is essential to take into account a certain kind of macroscopic nature of the total system. Some attempts at extracting the exponential decay law from quantum theory, aiming at the master equation, are briefly reviewed, including van Hove's pioneering work and his well-known ``$λ^2T$" limit. We clarify these general arguments by introducing and studying a solvable dynamical model. Some implications for the quantum measurement problem are also discussed, in particular in connection with dissipation.