A Two-Body Problem for a Class of Coupling Potentials with Harmonic Oscillator Interaction in Noncommutative Phase Space: Path Integral Approach

Abstract

Recently Gnatenko et al. (J Phys Stud 21:4002, 2017) have treated the two-body problem with harmonic oscillator interaction in noncommutative phase space. According to the path integral formulation, we generalize this process by adding the coupling potentials ( $$\kappa {\mathbf {P}} ^{\left( a\right) }\cdot {\mathbf {P}}^{\left( b\right) }$$ and $${\tilde{\kappa }} {\mathbf {X}}^{\left( a\right) }\cdot {\mathbf {X}}^{\left( b\right) }$$ ) together with a Harmonic oscillator interaction in noncommutative phase space. Moreover, in order to convert this problem into a one-particle system, we have added a third condition along with what this paper provided. As well as, when $$ \kappa $$ and $${\tilde{\kappa }}$$ equal to zero exact expressions for the energy levels of a two-particle system with harmonic oscillator interaction in noncommutative phase space have been found.