Solution of the quadratic system in non-commutative phase space (Coupled harmonic oscillator pair in the mixed non-commutative phase space)

Abstract

In this study, we focused on solving the Schr¨odinger equation for a pair of harmonic oscillators in a non-commutative phase space. Our primary goal was to understand the effect of phase space deformation on the system dynamics and derive analytical solutions for the modified Hamiltonian. We began with an extensive literature review on non-commutative phase space, which allowed us to redefine momentum operators and utilize modified Poisson brackets. These brackets express the impact of deformation in non-commutative geometry and quantum gravity contexts. We constructed a modified Hamiltonian using deformation parameters θ and σ, and a coupling constant λ, as corrections to the traditional Hamiltonian. We then solved the Schr¨odinger equation and derived an expression for the corrected frequency (Ω). Next, we conducted a numerical comparison between the Schr¨odinger and ¯ Feynman approaches for the same system under deformed and standard conditions. The results showed perfect agreement between the approaches when deformation parameters approach zero. Additionally, energy levels and probability density for the system under weak deformations exhibited similar behaviors for both approaches, depending on the deformation parameters. This highlights the role of deformation in generating noticeable differences in energy values and probability density.