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.

Fatih Erman

O. Teoman Turgut

American Journal of Physics

美国物理学杂志 (AJP) 的使命是发表有关物理学教育和文化方面的文章，这些文章对物理学学生、教育工作者和研究人员的不同受众来说是有用的、有趣的和可访问的。我们的读者通常会在他们的专业之外阅读，以扩大他们对物理学的理解，并在本科和研究生阶段扩展和增强他们的教学工具包。

The mission of the American Journal of Physics (AJP) is to publish articles related to physics education and culture that are useful, interesting, and accessible to a diverse audience of physics students, educators, and researchers. Our readers often read beyond their professional fields to broaden their understanding of physics and to expand and enhance their teaching toolkit at the undergraduate and graduate levels.

《American Journal of Physics》（美国物理学杂志，AJP）的使命是发表与物理学教育和文化相关的文章，这些文章对物理学学生、教育工作者和研究人员等不同受众具有实用性、趣味性和可读性。我们的读者通常会在专业领域之外进行阅读，以拓宽他们对物理学的理解，并在本科和研究生阶段扩展和丰富他们的教学工具。

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Completeness of energy eigenfunctions for the reflectionless potential in quantum mechanics

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