Algebraic Solution of the Harmonic Oscillator With Minimal Length Uncertainty Relations

TL;DR

This paper presents an algebraic method to solve the quantum harmonic oscillator with minimal length uncertainty, revealing a deformed SU(1,1) algebra structure and constructing eigenvalues and eigenstates without solving the Schrödinger equation directly.

Contribution

It introduces a novel algebraic approach to solve the harmonic oscillator under minimal length uncertainty relations, generalizable to higher dimensions.

Findings

Eigenvalues and eigenstates form an infinite-dimensional deformed SU(1,1) representation.

Method is independent of the Schrödinger equation solution.

Approach can be extended to D-dimensional oscillators with non-commuting coordinates.

Abstract

In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates are constructed algebraically and they form the infinite-dimensional representation of the deformed SU(1,1) algebra. Our construction is independent of prior knowledge of the exact solution of the Schr\"odinger equation of the model. The approach can be generalized to the $D$-dimensional oscillator with non-commuting coordinates.