Commutator Anomaly in Noncommutative Quantum Mechanics

TL;DR

This paper derives the anomaly in the commutator of physical observables in noncommutative quantum mechanics, explores uncertainty relations, and extends the Schrödinger equation to noncommutative phase space.

Contribution

It introduces a generalized Bopp's shift to formulate the Schrödinger equation and calculates the anomaly term in commutators within NCQM.

Findings

Derived the anomaly term of commutators in NCQM

Established uncertainty relations for various operators in NCQM

Extended Schrödinger equation to noncommutative phase space

Abstract

In this letter, firstly, the Schr$\ddot{o}$dinger 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.