Effect of noncommutativity on the spectrum of free particle and harmonic oscillator in rotationally invariant noncommutative phase space

TL;DR

This paper investigates how rotationally invariant noncommutativity influences the spectra of free particles and harmonic oscillators, revealing that noncommutativity induces discretization and modifies physical parameters like mass and frequency.

Contribution

It introduces a rotationally invariant noncommutative algebra and analyzes its effects on quantum spectra up to second order in noncommutativity parameters, providing new insights into noncommutative phase space.

Findings

Momentum noncommutativity makes the free particle spectrum discrete.

Noncommutativity alters the mass and frequency of the harmonic oscillator.

Eigenvalues of the squared length operator are derived, constraining length scales.

Abstract

We consider rotationally invariant noncommutative algebra with tensors of noncommutativity constructed with the help of additional coordinates and momenta. The algebra is equivalent to well known noncommutative algebra of canonical type. In the noncommutative phase space with rotational symmetry influence of noncommutativity on the spectrum of free particle and spectrum of harmonic oscillator is studied up to the second order in the parameters of noncommutativity. We find that because of momentum noncommutativity the spectrum of free particle is discrete and corresponds to the spectrum of harmonic oscillator in the ordinary space (space with commutative coordinates and commutative momenta). We obtain the spectrum of the harmonic oscillator in the rotationally invariant noncommutative phase space and conclude that noncommutativity of coordinates affects on its mass. The frequency of the oscillator is affected by the coordinate noncommutativity and the momentum noncommutativity. On the basis of the results, the eigenvalues of squared length operator are found and restrictions on the value of length in noncommutative phase space with rotational symmetry are analyzed.