# Potential Algebra Approach to Quantum Mechanics with Generalized Uncertainty Principle

## TL;DR

This paper explores the potential algebra approach to quantum mechanics with a generalized uncertainty principle, demonstrating solutions for specific models using Lie algebra representations, particularly $rak{su}(2)$, for systems like the harmonic and Dirac oscillators.

## Contribution

It introduces a method to solve quantum models with a minimal length uncertainty using $rak{su}(2)$ algebra representations, extending the potential algebra approach to these systems.

## Key findings

- Eigenvalue equations can be solved via $rak{su}(2)$ representations.
- The approach applies to both non-relativistic and relativistic oscillators.
- Solutions are expressed solely in terms of Lie algebra representations.

## Abstract

In this note, we study the potential algebra for several models arising out of quantum mechanics with generalized uncertainty principle. We first show that the eigenvalue equation corresponding to the momentum-space Hamiltonian \[H=-(1+\beta p^{2})\frac{d}{dp}(1+\beta p^{2})\frac{d}{dp}+g(g-1)\beta^{2}p^{2}-g\beta,\] which is associated with some one-dimensional models with minimal length uncertainty, can be solved by the unitary representations of the Lie algebra $\mathfrak{su}(2)$ if $g\in\{\tfrac{1}{2},1,\tfrac{3}{2},2,\cdots\}$. We then apply this result to spectral problems for the non-relativistic harmonic oscillator as well as the relativistic Dirac oscillator in the presence of a minimal length and show that these problems can be solved solely in terms of $\mathfrak{su}(2)$.