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

Abstract

We present the 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 criterion of the eigenfunctions in a non-deformed case is an emergent condition and the semiclassical solutions exactly coincide with the quantum mechanical results. The behavior of the wavefunctions at the origin in coordinate space and in quasiposition space is discussed finally.