Particle in uniform field in noncommutative space with preserved time reversal and rotational symmetries

TL;DR

This paper investigates a particle in a uniform field within a noncommutative space that preserves time reversal and rotational symmetries, deriving exact energy levels and wavefunctions, and analyzing how noncommutativity affects particle motion.

Contribution

It provides an exact solution for a particle in a uniform field in a symmetric noncommutative space, highlighting the effects of noncommutativity on particle dynamics.

Findings

Particle energy levels are exactly determined.

Motion along the field remains unchanged by noncommutativity.

Noncommutativity affects the particle's effective mass in perpendicular directions.

Abstract

Quantized space described by time reversal invariant and rotationally invariant noncommutative algebra of canonical type is studied. A particle in uniform field is considered. We find exactly the energy of a particle in uniform field in the quantized space and its wavefunctions. It is shown that the motion of the particle in the field direction in time reversal invariant and rotationally invariant noncommutative space is the same as in the ordinary space (space with the ordinary commutation relations for operators of coordinates and operators of momenta). Noncommutativity of coordinates has influence only on the motion of the particle in the directions perpendicular to the field direction. Namely, space quantization has effect on the mass of the particle.