The decay law can have an irregular character

Abstract

Within a well-known decay model describing a particle confined initially within a spherical δ potential shell, we consider the situation when the undecayed state has an unusual energy distribution decaying slowly as k → ∞; the simplest example corresponds to a wave function constant within the shell. We show that the non-decay probability as a function of time behaves then in a highly irregular, most likely fractal way.PACS numbers: 03.65.XpThe decay of an unstable quantum systems is one of the effects frequently discussed and various aspects of such processes were considered. To name just a few, recall the long-time deviation from the exponential decay law[1], the short-time behavior related to the Zeno and anti-Zeno effects [3,4,5,6], revival effects such the classical one in the kaon-antikaon system, etc. In all the existing literature [7], however, the decay law is treated as a smooth function, either explicitly or implicitly, e.g. by dealing with its derivatives. The aim of this letter is to show there are situation when it is not the case.A hint why it could be so comes from the behavior of Schrödinger wave functions during the time evolution. While in most cases the evolution causes smoothing [8], it may not be true for for a particle confined in a potential well and the initial state does not belong to the domain of the Hamiltonian. A simple and striking example was found by M. Berry [9] for a rectangular hardwall box, and independently by B. Thaller [10] for a onedimensional infinite potential well. It appears that if the initial wave function is constant, it evolves into a steplikeshaped ψ(x, t) for times which are rational multiples of the period, t = qT with q = N/M , and the number of steps increases with growing M , while for an irrational q the function ψ(x, t) is fractal w.r.t. the variable x.One can naturally ask what will happen if the hard wall is replaced by a semitransparent barrier through which the particle can tunnel into the outside space. In a broad sense this is one of the most classical decay model which can be traced back to [11]. We will deal with its particular case when the barrier is given by a spherical δ potential which is sometimes called Winter model being introduced for the first time, to our knowledge, in [12]; see also [13]. The described behavior of the wave function in the absence of tunneling suggests that in the decaying system the irregular time dependence could also be visible, both in the wave function and in various quantities derived from it [14], at least in the weak coupling case. The aim of the present letter is to demonstrate that this conjecture is indeed valid.To be concrete, we will study a spinless nonrelativistic quantum particle described by the Hamiltonianwith a fixed R > 0; we use rational units, = 2m = 1. For simplicity we restrict our attention to the s-wave part of the problem, writing thus the wave functions as ψ( r, t) = 1 √ 4π r −1 φ(r, t) with the associated Hamiltonianin the lowest partial wave. We are interested in the time evolution determine...

Solvable Models of Resonances and Decays

“…Using these relations and denoting k −n := −k n with v −n being the associated solution of the equation These explicit formulae allow us to find P ψ (t) numerically. Let us quote an example worked out in [EF07] in which R = 1 and α = 500; the initial wave function is chosen to be constant, i.e. the ground state of the Neumann Laplacian in L 2 (B R (0)) which corresponds to φ(r, 0) = R −3/2 √ 3r χ [0,R] (r) .…”

“…These explicit formulae allow us to find P ψ (t) numerically. Let us quote an example worked out in [EF07] in which R = 1 and α = 500; the initial wave function is chosen to be constant, i.e. the ground state of the Neumann Laplacian in…”

Time-of-arrival probabilities and quantum measurements. III. Decay of unstable states

“…Hence, for the exponential decay law to be valid for times much larger than the decay time Γ −1 , it is necessary that 16) or, defining the variation in transmission probability according to the momentum spread σ of the initial state…”

Decay Law Regularity

“…A numerical analysis of this and similar models [1] for a particular choice of ψ 0 gives a highly irregular function. A conjecture to be considered is that for ψ 0 (a−) = 0 the function P ψ0 (·) might be nowhere differentiable.…”