Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics

TL;DR

This paper provides an elementary proof of the completeness of energy eigenfunctions for the reflectionless potential in quantum mechanics, including derivations of eigenstates and demonstrations of fundamental properties.

Contribution

It offers a new, straightforward proof of completeness for reflectionless potentials and introduces an elegant derivation of normalized eigenstates using harmonic oscillator analogs.

Findings

Completeness of bound and scattering states is explicitly verified.

Wave functions for single bound states can be derived from continuum states.

Parity eigenstates are used to demonstrate completeness explicitly.

Abstract

There are 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.