Modified quantum current density under minimal uncertainty in momentum and nonlocal potential within noncommutative Framework

In this paper, we investigate the Schrödinger and continuity equations in the presence of both local and nonlocal potentials, the latter arising from electron–electron interactions, within the framework of minimal momentum uncertainty in both commutative and noncommutative phase–spaces. In particular, a Frahn–Lemmer-type nonlocal potential is employed. The combined effects of phase–space noncommutativity and potential nonlocality on the quantum density and current density are thoroughly analyzed. It is found that the conventional definition of current density fails to satisfy the current conservation condition under these generalized settings. Therefore, a modified definition of current density that properly accounts for these contributions is proposed. We subsequently demonstrate that the current obtained from this new formulation satisfies the continuity equation. Additionally, numerical results are presented to extensively explore the behavior of the system under various conditions. Phase–space noncommutativity is introduced via both the Moyal Formula: see textproduct and the linear Bopp-shift transformation.