Comparing exact energy solutions of quartic eigenvalue polynomials in commutative, non-commutative and non-commutative phase frameworks for boson pi(-)

摘要

In this paper, the behavior of a Duffin-Kemmer-Petiau (DKP) boson particle in the presence of a harmonic energy-dependent interaction, under the influence of an external magnetic field is precisely studied. In order to exactly solve all equations in commutative (C), non-commutative (NC) and non-commutative phase (NCP) frameworks, the Nikiforov-Uvarov (NU) powerful exact approach is employed. All these attempts end up with solving their quartic equations, trying to find and discuss on their discriminant function Delta, in a unique way which has never been discussed for any boson in any other research, especially for the boson pi(-) on which, we have been exclusively concerned. We finally succeeded to obtain the exact energy spectrums and wave functions under the effects of NC and NCP parameters and energy-dependent interaction on energy eigenvalues. In this step, we analyze the behaviors of their quartic energy eigenvalue polynomials in three sections and accurately compare all achieved physical-admissible roots one by one. This comparison surprisingly shows that the NC and NCP effects on the other hand, and the assumed harmonic energy-dependent interaction on the other hand, have almost the same order of perturbation effects for limited amounts of the magnetic field in a system of DKP bosons. Furthermore, through some calculations within this paper, we came up with a very crucial point about the NU method which was mistakenly being used in many papers by several researchers and improved it to be used safely.