Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics

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Bibliographic Details
Main Authors:	Erman, F., Turgut, O. T.
Format:	Preprint
Published:	2024
Subjects:	Quantum Physics Mathematical Physics
Online Access:	Acceder al recurso
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author	Erman, F. Turgut, O. T.
author_facet	Erman, F. Turgut, O. T.
contents	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.
format	Preprint
id	arxiv_https___arxiv_org_abs_2411_14941
institution	arXiv
publishDate	2024
record_format	arxiv
spellingShingle	Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics Erman, F. Turgut, O. T. Quantum Physics Mathematical Physics 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.
title	Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics
topic	Quantum Physics Mathematical Physics
url	https://arxiv.org/abs/2411.14941

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