One-dimensional hydrogen atom with minimal length uncertainty and maximal momentum

TL;DR

This paper derives exact energy levels and wavefunctions for a one-dimensional hydrogen atom within a GUP framework that incorporates minimal length and maximal momentum, aligning with quantum gravity theories.

Contribution

It provides the first exact solutions for the hydrogen atom under a GUP that includes both minimal length and maximal momentum constraints.

Findings

Exact energy eigenvalues match semiclassical results.

Eigenfunctions are explicitly derived.

Supports the consistency of GUP with quantum gravity theories.

Abstract

We present exact energy eigenvalues and eigenfunctions of the one-dimensional hydrogen atom in the framework of the Generalized (Gravitational) Uncertainty Principle (GUP). This form of GUP is consistent with various theories of quantum gravity such as string theory, loop quantum gravity, black-hole physics, and doubly special relativity and implies a minimal length uncertainty and a maximal momentum. We show that the quantized energy spectrum exactly agrees with the semiclassical results.