Time evolution of a decaying quantum state: Evaluation and onset of nonexponential decay

Abstract

The method developed by van Dijk, Nogami, and Toyama for obtaining the time-evolved wave function of a decaying quantum system is generalized to potentials and initial wave functions of noncompact support. The long time asymptotic behavior is extracted and employed to predict the timescale for the onset of nonexponential decay. The method is illustrated with a Gaussian initial wave function leaking through Eckart's potential barrier on the half-line.