Completeness and orthonormality of the energy eigenfunctions of the Dirac delta derivative potential

Abstract

The orthonormality and completeness of the energy eigenfunctions solution for the Schrödinger equation are usually hard to prove explicitly. There are few examples treated in the literature that give an explicit demonstration of these properties. In this work, we are going to orthonormalize the energy eigenfunctions of the solutions of the Schrödinger equation with contact potential given by the Dirac delta derivative potential λδ′(x), where λ is the coupling constant and prove that they are complete. The solutions for this problem include both continuum and discrete energy spectra. This makes it a perfect choice in the sense that it covers all possible cases of the solutions in quantum mechanics. The bound states solutions are normalized as usual, while the scattering states are normalized in continuum sense. The scattering solutions are shown to be orthogonal to each other and to the bound states. Using these orthonormal eigenfunctions an explicit demonstration of the completeness is given. It is found that both the bound and scattering solutions should be considered to ensure completeness. The calculation shows that the scattering state solutions are complete by themselves except for a certain value of the energy eigenvalue. This energy eigenvalue was shown to be precisely consistent with the energy eigenvalue of the bound state.