Generalized Uncertainty Relation in the Non-commutative Quantum Mechanics

Abstract

In this paper the non-commutative quantum mechanics (NCQM) with the generalized uncertainty relations is discussed. Four each uncertainty relation, wave functions saturating each uncertainty relation are explicitly constructed. The unitary operators relating the non-commutative position and momentum operators to the commutative position and momentum operators are also investigated. We also discuss the uncertainty relation related to the harmonic oscillator.