Completeness of energy eigenfunctions for the reflectionless potential in quantum mechanics.

There are a few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians. Editor's Note: The reflectionless potential, also called the Pöschl–Teller potential, is one of the less-often discussed quantum potentials with known analytical solutions. The system has many interesting properties, including the fact that it allows both bound and scattering states and that the potential well never reflects incident particles. The authors here present new ways to think about the somewhat counter-intuitive completeness of the set of bound and scattering states of the system. [ABSTRACT FROM AUTHOR]