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Modern Physics Letters A

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

Ilyas Haouam

Published Feb 11, 2026• 0 citations
Publisher

Abstract

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 text]product and the linear Bopp-shift transformation.

DOI
10.1142/s0217732326500458

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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 text]product and the linear Bopp-shift transformation.
We studied the continuity equation in presence of a local potential, and a non-local potential arising from electron-electron interaction in both commutative and non-commutative phase-space. Furthermore, we examined the influence of the phase-space non-commutativity on both the locality and the non-locality, where the definition of current density in commutative phase-space cannot satisfy the condition of current conservation, but with the steady state, in order to solve this problem, we give a new definition of the current density including the contribution due to the non-local potential. We showed that the calculated current based on the new definition of current density maintains the current. As well for the case when the non- commutativity in phase-space considered, we found that the conservation of the current density completely violated; and the non-commutativity is not suitable for describing the current density in presence of non-local and local potentials. Nevertheless, under some conditions, we modified the current density to solve this problem. Subsequently, as an application we studied the Frahn-Lemmer non-local potential, taking into account that the employed methods concerning the phase-space non-commutativity are both of Bopp-shift linear transformation through the Heisenberg-like commutation relations, and the Moyal-Weyl product.
In this letter, firstly, the Schrddoto\\ddot{o}ddotodinger equation on noncommutative phase space is given by using a generalized Bopp's shift. Then the anomaly term of commutator of arbitrary physical observable operators on noncommutative phase space is obtained. Finally, the basic uncertainty relations for space-space and space-momentum as well as momentum-momentum operators in noncommutative quantum mechanics (NCQM), and uncertainty relation for arbitrary physical observable operators in NCQM are discussed.
The continuity equation fixes only the divergence of the quantum probability current, leaving the flow of probability underdetermined. We show that Galilean boost covariance, imposed pointwise at fixed time, nearly resolves this ambiguity. For any one-body current, local or nonlocal, pure, mixed, or many-body, the boost law holds exactly when the current equals the Schrödinger current plus a linear functional of the density. Equivalently, the current is covariant under local phase changes, or its density and momentum density satisfy the canonical commutator of the nonrelativistic current algebra. Among translation-invariant currents, rotational covariance together with conservation removes the remaining density term in three or more dimensions, while conservation together with the momentum–translation relation removes it in every dimension. The same framework classifies known alternatives: Nelson and Doebner–Goldin currents obey the algebra but are not conserved; chiral currents are conserved but violate the algebra; convolution kernels violate the boost law; and a conserved boost-covariant current exists exactly when the fff-sum rule holds.
In this paper, we obtained the three-dimensional Pauli equation for a spin-1/2 particle in the presence of an electromagnetic field in a noncommutative phase-space as well as the corresponding deformed continuity equation, where the cases of a constant and non-constant magnetic fields are considered. Due to the absence of the current magnetization term in the deformed continuity equation as expected, we had to extract it from the noncommutative Pauli equation itself without modifying the continuity equation. It is shown that the non-constant magnetic field lifts the order of the noncommutativity parameter in both the Pauli equation and the corresponding continuity equation. However, we successfully examined the effect of the noncommutativity on the current density and the magnetization current. By using a classical treatment, we derived the semi-classical noncommutative partition function of the three-dimensional Pauli system of the one-particle and N-particle systems. Then, we employed it for calculating the corresponding Helmholtz free energy followed by the magnetization and the magnetic susceptibility of electrons in both commutative and noncommutative phase-spaces. Knowing that with both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product, we introduced the phase-space noncommutativity in the problems in question.
Schrödinger equation with given, a priori known current is formulated. A nonzero current density is maintained in the quantum system via a subsidiary condition imposed by vector, local Lagrange multiplier. Constrained minimization of the total energy on the manifold of an arbitrary current density topology results into a nonlinear self-consistent Schrödinger equation. The applications to electronic transport in two-terminal molecular devices are developed and new macroscopic definition of a molecular current–voltage characteristic is proposed. The Landauer formula for the conductance of an ideal one-dimensional lead is obtained within the approach. The method is examined by modeling of current carrying states of one-dimensional harmonic oscillator.