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Generalized Uncertainty Relation in the Non-commutative Quantum Mechanics

International Journal of Theoretical Physics volume 55, pages 2989–3000 (2016)Cite this article

Abstract

In this paper the non-commutative quantum mechanics (NCQM) with the generalized uncertainty relations \({\Delta } x_{1} {\Delta } x_{2} \ge \frac {\theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac {\bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar _{eff}}{2}\) 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.

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Acknowledgments

I acknowledge to the reviewer for helpful comment. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2015R1D1A1A01057792).

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  1. Department of Physics and Research Institute of Natural Science, College of Natural Science, Gyeongsang National University, Jinju, 660-701, Korea

    Won Sang Chung

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  1. Won Sang Chung
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Correspondence to Won Sang Chung.

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Chung, W.S. Generalized Uncertainty Relation in the Non-commutative Quantum Mechanics. Int J Theor Phys 55, 2989–3000 (2016). https://doi.org/10.1007/s10773-016-2931-0

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  • DOI: https://doi.org/10.1007/s10773-016-2931-0

Keywords

  • Non-commutative quantum mechanics
  • Generalized uncertainty relation unitary operator