Position-dependent noncommutative quantum models: Exact solution of the harmonic oscillator

TL;DR

This paper derives an exact analytical solution for the harmonic oscillator in a four-dimensional, position-dependent noncommutative phase space with specific commutation relations, expanding understanding of quantum models with variable noncommutativity.

Contribution

It provides the first exact solution of the harmonic oscillator in a position-dependent noncommutative phase space with specific algebraic relations.

Findings

Exact eigenvalues and eigenstates derived for the deformed harmonic oscillator.

Analytical method developed for solving quantum systems in position-dependent noncommutative spaces.

Enhanced understanding of quantum behavior in variable noncommutative geometries.

Abstract

This paper is devoted to find the exact solution of the harmonic oscillator in a position-dependent 4-dimensional noncommutative phase space. The noncommutative phase space that we consider is described by the commutation relations between coordinates and momenta: $[\hat{x}^1,\hat{x}^2]=i\theta(1+\omega_2 \hat x^2)$, $[\hat{p}^1,\hat{p}^2]=i\bar\theta$, $[\hat{x}^i,\hat{p}^j]=i\hbar_{eff}\delta^{ij}$. We give an analytical method to solve the eigenvalue problem of the harmonic oscillator within this deformation algebra.