A note on the one-dimensional hydrogen atom with minimal length uncertainty

TL;DR

This paper derives exact energy spectra and eigenfunctions for a one-dimensional hydrogen atom considering minimal length uncertainty, revealing how quantum conditions adapt and semiclassical solutions align with quantum results.

Contribution

It provides a complete analysis of the hydrogen atom with minimal length, including quantization conditions and wave function behavior, extending traditional quantum mechanics.

Findings

Exact energy spectrum and eigenfunctions derived

Semiclassical solutions match quantum results

Wave function behavior at the origin analyzed

Abstract

We present exact energy spectrum and eigenfunctions of the one-dimensional hydrogen atom in the presence of the minimal length uncertainty. By requiring the self-adjointness property of the Hamiltonian, we completely determine the quantization condition. We indicate that the single-valuedness criteria of the eigenfunctions in non-deformed case is an emergent condition and the semiclassical solutions exactly coincide with the quantum mechanical results. The behavior of the wave functions at the origin in coordinate space and in quasiposition space is discussed finally.