Spectra of Harmonic Oscillators with GUP and Extra Dimensions


In this paper, we address the spectra of simple harmonic oscillators based on the generalized uncertainty principle (GUP) with a Kaluza-Klein compactified extra dimension. We show that in this scenario, to make the results compatible with experiments, the minimal length scale equals to the radius of compact extra dimension.

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Mu, B. , Yu, R. and Wang, D. (2019) Spectra of Harmonic Oscillators with GUP and Extra Dimensions. Journal of High Energy Physics, Gravitation and Cosmology, 5, 279-290. doi: 10.4236/jhepgc.2019.51015.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.


[1] Veneziano, G. (1986) A Stringy Nature Needs Just Two Constants. Europhysics Letters, 2, 199.
[2] Gross, D.J. and Mende, P.F. (1988) String Theory beyond the Planck Scale. Nuclear Physics B, 303, 407-454.
[3] Amati, D., Ciafaloni, M. and Veneziano, G. (1989) Can Spacetime Be Probed below the String Size? Physics Letters B, 216, 41-47.
[4] Konishi, K., Paffuti, G. and Provero, P. (1990) Minimum Physical Length and the Generalized Uncertainty Principle in String Theory. Physics Letters B, 234, 276-284.
[5] Guida, R., Konishi, K. and Provero, P. (1991) On the Short Distance Behavior of String Theories. Modern Physics Letters A, 6, 1487-1503.
[6] Maggiore, M. (1993) A Generalized Uncertainty Principle in Quantum Gravity. Physics Letters B, 304, 65-69.
[7] Zhao, R., Zhang, L.C., Wu, Y.Q., et al. (2010) Generalized Uncertainty Principle and Tunneling Radiation of the SAdS5 Black Hole. Chinese Physics B, 19, 010402.
[8] Corda, C., Chakraborty, S. and Saha, S. (2015) Light from Black Holes and Uncertainty in Quantum Gravity. Electronic Journal of Theoretical Physics, 12, 107.
[9] Haldar, S., Corda, C. and Chakraborty, S. (2018) Tunnelling Mechanism in Noncommutative Space with Generalized Uncertainty Principle and Bohr-Like Black Hole. Advances in High Energy Physics, 2018, Article ID: 9851598.
[10] Mu, B., Wang, P. and Yang, H. (2015) Minimal Length Effects on Tunnelling from Spherically Symmetric Black Holes. Advances in High Energy Physics, 2015, Article ID: 898916.
[11] Mu, B.R., Wang, P. and Yang, H.T. (2015) Minimal Length Effects on Schwinger Mechanism. Communications in Theoretical Physics, 63, 715.
[12] Gecim, G. and Sucu, Y. (2018) Quantum Gravity Effect on the Hawking Radiation of Charged Rotating BTZ Black Hole. General Relativity and Gravitation, 50, 152.
[13] Kempf, A., Mangano, G. and Mann, R.B. (1995) Hilbert Space Representation of the Minimal Length Uncertainty Relation. Physical Review D, 52, 1108.
[14] Kempf, A. (1997) Non-Pointlike Particles in Harmonic Oscillators. Journal of Physics A, 30, 2093.
[15] Brau, F. (1999) Minimal Length Uncertainty Relation and Hydrogen Atom. Journal of Physics A, 32, 7691-7696.
[16] Brau, F. and Buisseret, F. (2006) Minimal Length Uncertainty Relation and Gravitational Quantum Well. Physical Review D, 74, Article ID: 036002.
[17] Chang, L.N., Minic, D., Okamura, N. and Takeuchi, T. (2002) Exact Solution of the Harmonic Oscillator in Arbitrary Dimensions with Minimal Length Uncertainty Relations. Physical Review D, 65, Article ID: 125027.
[18] Dadic, I., Jonke, L. and Meljanac, S. (2003) Physical Review D, 67, Article ID: 087701.
[19] Nozari, K. and Azizi, T. (2006) Some Aspects of Gravitational Quantum Mechanics. General Relativity and Gravitation, 38, 735-742.
[20] Stetsko, M.M. and Tkachuk, V.M. (2006) Perturbation Analysis for Competing Reactions with Initially Separated Components. Physical Review A, 74, Article ID: 012101.
[21] Stetsko, M.M. (2006) Corrections to the ns-Levels of Hydrogen Atom in Deformed Space with Minimal Length. Physical Review A, 74, Article ID: 062105.
[22] Benczik, S.Z. (2007) Investigations on the Minimal-Length Uncertainty Relation. PhD Thesis, Virginia Polytechnic Institute and State University, Blacksburg.
[23] Battisti, M.V. and Montani, G. (2007) The Big-Bang Singularity in the Framework of a Generalized Uncertainty Principle. Physics Letters B, 656, 96-101.
[24] Battisti, M.V. and Montani, G. (2008) Quantum Dynamics of the Taub Universe in a Generalized Uncertainty Principle Framework. Physical Review D, 77, Article ID: 023518.
[25] Das, S. and Vagenas, E.C. (2008) Universality of Quantum Gravity Corrections. Physical Review Letters, 101, Article ID: 221301.
[26] Mu, B., Wu, H. and Yang, H. (2011) The Generalized Uncertainty Principle in the Presence of Extra Dimensions. Chinese Physics Letters, 28, Article ID: 091101.
[27] Antoniadis, I. (1990) A Possible New Dimension at a Few TeV. Physics Letters B, 246, 377-384.
[28] Arkani-Hamed, N., Dimopoulos, S. and Dvali, G.R. (1998) The Hierarchy Problem and New Dimensions at a Millimeter. Physics Letters B, 429, 263-272.
[29] Antoniadis, I., Arkani-Hamed, N., Dimopoulos, S. and Dvali, G.R. (1998) New Dimensions at a Millimeter to a Fermi and Superstrings at a TeV. Physics Letters B, 436, 257-263.
[30] Randall, L. and Sundrum, R. (1999) A Large Mass Hierarchy from a Small Extra Dimension. Physical Review Letters, 83, 3370-3373.
[31] Randall, L. and Sundrum, R. (1999) An Alternative to Compactification. Physical Review Letters, 83, 4690-4693.
[32] Dvali, G.R., Gabadadze, G. and Porrati, M. (2000) 4D Gravity on a Brane in 5D Minkowski Space. Physics Letters B, 485, 208-214.
[33] Appelquist, T., Cheng, H.C. and Dobrescu, B.A. (2001) Bounds on Universal Extra Dimensions. Physical Review D, 64, Article ID: 035002.
[34] Cremades, D., Ibanez, L.E. and Marchesano, F. (2002) Standard Model at Intersecting D5-Branes: Lowering the String Scale. Nuclear Physics B, 643, 93-130.
[35] Kokorelis, C. (2004) Exact Standard Model Structures from Intersecting D5-Branes. Nuclear Physics B, 677, 115-163.
[36] Li, Z.G. and Ni, W.T. (2008) Extra Dimensions and Atomic Transition Frequencies. Chinese Physics B, 17, 70-75.
[37] Bezerra, V.B. and Rego-Monteiro, M.A. (2004) Some Boundary Effects in Quantum Field Theory. Physical Review D, 70, Article ID: 065018.
[38] Takenaga, K. (2000) Quantization Ambiguity and Supersymmetric Ground State Wave Functions. Physical Review D, 62, Article ID: 065001.
[39] Doncheski, M.A. and Robinett, R.W. (2003) Wave Packet Revivals and the Energy Eigenvalue Spectrum of the Quantum Pendulum. Annals of Physics, 308, 578-598.
[40] McLachlan, N.W. (1947) Theory and Application of Mathieu Functions. Clarendon Press, Oxford.
[41] Hradil, et al. (2006) Minimum Uncertainty Measurements of Angle and Angular Momentum. Physical Review Letters, 97, Article ID: 243601.
[42] Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover Publications, New York, 724.

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