Nonexponential decay law

Abstract

The decay of a nonstationary state usually starts as a quadratic function of time and ends as an inverse power law (possibly with oscillations). Between these two extremes, the familiar exponential decay law may be approximately valid. The main purpose of this paper is to find the conditions which must be satisfied by the Hamiltonian and by the initial state, for the exponential law to have a significant domain of validity. It is shown that the evolution of a nonstationary state is governed by a nonnegative function W( E), having the dimensions of an energy. Among its properties are: the energy uncertainty is given by (ΔH) 2 = ƒW(E)dE , and the inverse lifetime by Γ = 2 πW( E 0), where E 0 is the expectation value of H. The detailed shape of W( E) defines two characteristic times between which the exponential decay law is a good approximation: roughly speaking, the smoother W( E), the larger the domain of validity of the exponential law. For instance, if W( E) is very smooth ( | dW dE | ⪡ 1 ) except for a sharp threshold at E = E thr, the transition from quadratic to exponential decay occurs for t ⋍ 1 (E 0 − E thr ) , and the transition from exponential to inverse power law when Γt ⋍ log[ (E 0 − E thr ) Γ ] .