Volume 136, number 6 PHYSICS LETTERS A lOApril 1989 NUMERICAL STUDY OF ZEN0 AND ANTI-ZEN0 EFFECTS IN A LOCAL POTENTIAL MODEL W.C. SCHIEVE Ilya Prigogine Centerfor Studies in Statistical.Mechanics, University of Texas at Austin. Austin, TX 78712, USA L.P. HORWITZ School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel and Jacob LEVITAN Department of Physics, Bar Ilan University, Ramat Gan, Israel Received 15 November 1988; accepted for publication 16 February 1989 Communicated by J.P. Vigier The effect of perturbation of a barrier on the passage of a wave packet is studied as an example of a perturbed unstable system. It is found that with a small periodic perturbation, significant modification of the effective lifetime occurs. The decay may be retarded (Zeno effect) or enhanced according to the relative phase of the perturbation and arrival time of the packet. In 1930, Wigner and Weisskopf [ 1 ] #’ formulated a quantum mechanical description of unstable sys- tems by defining the probability amplitude for the persistence of the unstable state as A(t)=(~ole-‘H’lWO). (1) They showed, using second order time dependent perturbation theory that one obtains in this way (for times neither too short nor too long) a formula for the line width associated with exponential decay. For very short times, it follows from eq. ( 1) (when HvO exists) that the decay rate vanishes at zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA t = 0; the prob- ability of persistence (often called the survival prob- ability) then evolves at most quadratically in time. Under certain conditions, the survival probability goes smoothly to the exponential mode, but there may, under other conditions, be oscillations (ac- cording to the behavior of the spectral function) [ 3 1. Chiu, Sudarshan and Misra [ 41, using a rather gen- lr’ For a rather complete review and discussion of the theory of unstable systems, see ref. [ 21. era1 model for such systems, have estimated that the onset for the exponential mode should occur for (2) where Ei is the energy of the initial state of the un- stable system and Et,, is the threshold of the contin- uous spectrum of the decay products. For very long times, it follows from the Paley-Wiener theorem that, for H bounded below, the survival probability de- creases to zero less rapidly than any exponential function. If the system is disturbed during the decay, for ex- ample by a measurement which selects the decay products, the remaining systems in the beam must be treated as in the intial state, and start their evo- lution again from the time of measurement. If such measurements are made with sufficiently high fre- quency > 1 IT,, the effective decay rate could, in principle, be significantly modified. In the limit of very high frequency measurements, the decay might be completely suppressed. This has been called the 264 0375-9601/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)