Position-Dependent Noncommutative Quantum Models: Exact Solution of the Harmonic Oscillator

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: [ˆx1, ˆx2] = iθ(1 + ω2ˆx2), [ˆp1, ˆp2] = i¯θ, [ˆxi, ˆpj] = ieff δij . We give an analytical method to solve the eigenvalue problem of the harmonic oscillator within this deformation algebra.