Probing Uncertainty Relations in Non-Commutative Space

TL;DR

This paper investigates uncertainty relations in non-commutative space, deriving improved bounds and exploring differences from standard quantum mechanics, including reverse relations and distinctions between linear and non-linear models.

Contribution

It introduces new uncertainty bounds in non-commutative space and analyzes differences between linear and non-linear harmonic oscillator models.

Findings

Derived better lower bounds for uncertainty relations in non-commutative space.

Established reverse uncertainty relations for incompatible variables.

Identified distinctions between Schrödinger and Heisenberg uncertainties in non-linear models.

Abstract

In this paper, we compute uncertainty relations for non-commutative space and obtain a better lower bound than the standard one obtained from Heisenberg's uncertainty relation. We also derive the reverse uncertainty relation for product and sum of uncertainties of two incompatible variables for one linear and another non-linear model of the harmonic oscillator. The non-linear model in non-commutating space yields two different expressions for Schr\"odinger and Heisenberg uncertainty relation. This distinction does not arise in commutative space, and even in the linear model of non-commutative space.