Novel Expression for the Wave Function of a Decaying Quantum System

Abstract

We report on a novel, practical method of determining the nonrelativistic wave function of a decaying quantum system at all positions and times. The system consists of a particle which is initially confined around the origin and leaks out tunneling through a potential barrier. We show that the wave function $\ensuremath{\psi}(r,t)$ of the particle can be expressed as a linear combination of Moshinsky functions, each of which is associated with a pole of the scattering $\mathbf{S}$ matrix of the system. The $\ensuremath{\psi}(r,t)$ so obtained is normalizable. In the light of this approach we reexamine the Gamow wave function, which is not normalizable. We elucidate the source of this difficulty.