Examining the completeness of bound and scattering states

Abstract

In quantum mechanics, the completeness of stationary states, including both bound and scattering states, is a fundamental but often neglected topic. This work tests the completeness of the stationary states of the one-dimensional delta-function well potential. A basis of left- and right- or even and odd stationary scattering states is furnished along with the stationary bound state. We derive their orthonormality conditions and test their completeness on two classes of states (1) a one-parameter family of non-stationary bound states, and (2) a two-parameter family of non-stationary bound states, which are regulated scattering states. Using expansion coefficients, completeness is verified, and the time evolution is determined. The transition from bound to scattering state is explored using (2). The work highlights the complexities involved in proving completeness rigorously, even in a simple quantum system, and suggests future investigations into more intricate systems.