Space-time evolution of a decaying quantum state

We examine a one-dimensional model of a decaying quantum system, e.g., one that could simulate alpha decay. A particle is initially confined in a regi on and leaks out tunnelling through a potential barrier. The same model has been examined before, primarily to study decay properties which are determ ined by the wavefunction of the particle within the potential barrier. Howe ver, the wavefunction of this model land of any similar models) outside the potential barrier has not been obtained explicitly as a function of positi on and time. We obtain the wavefunction outside as well as inside the poten tial barrier by serving the time-dependent Schrodinger equation, and explor e various features of the space-time development of the system. In an early study by Winter it was pointed out that the probability current just outsi de the barrier after a long time, may fluctuate and can be negative, i.e., inward. We find that under certain circumstances the amplitude of the fluct uations increases significantly as the distance from the barrier increases. We propose a simple approximate wavefunctiqn that works well when the deca y is very slow The Gamow wavefunction, commonly used to describe the alpha- decay process, is not appropriate at large distances. Our wavefunction can be used anywhere and is normalized in the usual manner.