# A new higher order GUP: one dimensional quantum system

## Abstract

In this paper we present a new higher order generalized (gravitational) uncertainty principle (GUP*) which has the maximal momentum as well as the minimal length. We discuss the position representation and momentum representation. We also discuss the position eigenfunction and maximal localization states. As examples we discuss one dimensional box problem and harmonic oscillator problem.

## 1 Introduction

The Eq. (7) involves the even terms in *P* only. In one dimension, the magnitude of the momentum is not *P* but \(|P|= \sqrt{ P^2}\). Thus, if we are to construct the new higher order GUP, we should use |*P*| instead of *P*, which makes us find the simpler form than the Eq. (7). In this paper we present a new higher order generalized (gravitational) uncertainty principle (GUP*) in the form \([X, P] = i\hbar /(1 -\beta { |P|} )\) which has the maximal momentum as well as the minimal length.

This paper is organized as follows: In Sect. 2 we present a new higher order generalized (gravitational) uncertainty principle. In Sect. 3 we discuss the position representation and momentum representation. In Sect. 4 we discuss the position eigenfunction and maximal localization states In Sect. 5 we discuss the one dimensional box problem. In Sect. 6 we discussed the harmonic oscillator problem.

## 2 A new higher order GUP

## 3 Representation

Now let us find two representations for the GUP*.

### 3.1 Momentum representation

*A*, the expectation value of

*A*for the wave function \(\Phi (p)\) is

### 3.2 Position representation

*A*, the expectation value of

*A*for the wave function \(\psi (x)\) is

## 4 Position eigenfunction and maximal localization states

## 5 One dimensional Box problem

*m*confined to the following one-dimensional box

## 6 Harmonic oscillator problem

### 6.1 The semiclassical solution

### 6.2 Perturbative solution

*n*’s, we have

*n*for \(\beta =0\) (brown) and \(\beta =0.2\) (pink).

### 6.3 Comparison with the semiclassical solution and perturbative solution

*n*’s for the ordinary case, semiclassical solution and perturbative solution. Table 1 shows the \(2E_n/\hbar w\) with \(n=0, 1, 2, 3, 4\) for the ordinary case, semiclassical solution and perturbative solution for the dimensionless quantity \(\beta \sqrt{ \hbar mw } =0.01\). This shows that the energy for the perturbative solution is larger than the semiclassical solution, which results from the fact that we considered only classical region in the semiclassical solution.

\( 2E_n/\hbar w \) for \( n =0,1, 2, 3, 4\)

| \( 2E_n^{\beta =0} /\hbar w \) | \( 2E_n^{SC} /\hbar w \) | \( 2E_n /\hbar w \) |
---|---|---|---|

0 | 1 | 1.0042 | 1.0113 |

1 | 3 | 3.0221 | 3.0226 |

2 | 5 | 5.0475 | 5.0480 |

3 | 7 | 7.0786 | 7.0790 |

4 | 9 | 9.1146 | 9.1150 |

### 6.4 Classical solution

## 7 Conclusion

We presented a new higher order generalized (gravitational) uncertainty principle (GUP*) in the form \([X, P] = i/(1 -\beta { |P|} )\) where \( { |P|} = \sqrt{ |P^2|}\). For GUP* we explicitly showed that it gives the minimal length. We also obtained the position representation and momentum representation. We discussed the position eigenfunction and Maximal localization states From the comparison with Ref. [14] and Refs. [20, 21], we found that the expectation values of the kinetic energy operator for both position eigenfunction and the maximal localization states is the smallest for GUP*. Like the Refs. [20, 21] the GUP* was shown to have the nonzero minimal wavelength. We discussed one dimensional box problem where we found that the energy increases due to GUP* effect. We obtained the semiclassical solution and perturbative solution for the harmonic oscillator problem numerically. We found that the energy for the perturbative solution is larger than the semiclassical solution, which resulted from the fact that we considered only classical region in the semiclassical solution.

## Notes

### Acknowledgements

It is a great pleasure for the authors to thank the referee for helpful comments.This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2015R1D1A1A01057792) and by Development Fund Foundation, Gyeongsang National University, 2018.

## References

- 1.H.S. Snyder, Phys. Rev.
**71**, 38 (1947)ADSCrossRefGoogle Scholar - 2.C.N. Yang, Phys. Rev.
**72**, 874 (1947)ADSCrossRefGoogle Scholar - 3.C.A. Mead, Phys. Rev. B
**135**, 849 (1964)ADSCrossRefGoogle Scholar - 4.F. Karolyhazy, Nuovo Cim. A
**42**, 390 (1966)ADSCrossRefGoogle Scholar - 5.G. Veneziano, A stringy nature needs just two constants. Europhys. Lett.
**2**, 199 (1986)ADSCrossRefGoogle Scholar - 6.D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B
**197**, 81 (1987)ADSCrossRefGoogle Scholar - 7.D.J. Gross, P.F. Mende, Phys. Lett. B
**197**, 129 (1987)ADSMathSciNetCrossRefGoogle Scholar - 8.D.J. Gross, P.F. Mende, String theory beyond the Planck scale. Nucl. Phys. B
**303**, 407 (1988)ADSMathSciNetCrossRefGoogle Scholar - 9.D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B
**216**, 41 (1989)ADSCrossRefGoogle Scholar - 10.K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B
**234**, 276 (1990)ADSMathSciNetCrossRefGoogle Scholar - 11.G. Veneziano, Quantum gravity near the Planck scale. In: Proceedings of PASCOS 90, Boston, p. 486 (1990) (unpublished)Google Scholar
- 12.M. Maggiore, Phys. Lett. B
**304**, 65 (1993)ADSMathSciNetCrossRefGoogle Scholar - 13.A. Kempf, Uncertainty relation in quantum mechanics with quantum group symmetry. J. Math. Phys.
**35**, 4483 (1994). hep-th/9311147ADSMathSciNetCrossRefGoogle Scholar - 14.A. Kempf, G. Mangano, R.B. Mann, Phys. Rev. D
**52**, 1108 (1995)ADSMathSciNetCrossRefGoogle Scholar - 15.A. Kempf, On quantum field theory with nonzero minimal uncertainties in positions and momenta. J. Math. Phys.
**38**, 1347–1372 (1997). hep-th/9602085ADSMathSciNetCrossRefGoogle Scholar - 16.F. Scardigli, Phys. Lett. B
**452**, 39 (1999)ADSCrossRefGoogle Scholar - 17.R.J. Adler, D.I. Santiago, Mod. Phys. Lett. A
**14**, 1371 (1999)ADSCrossRefGoogle Scholar - 18.S. Capozziello, G. Lambiase, G. Scarpetta, Int. J. Theor. Phys.
**39**, 15 (2000)CrossRefGoogle Scholar - 19.F. Scardigli, R. Casadio, Class. Quantum Gravity
**20**, 3915 (2003)ADSCrossRefGoogle Scholar - 20.P. Pedram, Phys. Lett. B
**714**, 317 (2012)ADSCrossRefGoogle Scholar - 21.P. Pedram, Phys. Lett. B
**718**, 638 (2012)ADSCrossRefGoogle Scholar - 22.M. Bojowald, A. Kempf, Phys. Rev. D
**86**, 085017 (2012)ADSCrossRefGoogle Scholar - 23.P. Pedram, Int. J. Mod. Phys. D
**19**, 2003–2009 (2010)ADSMathSciNetCrossRefGoogle Scholar - 24.K. Nozari, P. Pedram, EPL
**92**, 50013 (2010)ADSCrossRefGoogle Scholar - 25.K. Nozari, M. Moafi, F. Rezaee Balef, Adv. High Energy Phys.
**2013**, 252178 (2013)Google Scholar - 26.K. Nozari, A. Etemadi, Phys. Rev. D
**85**, 104029 (2012)ADSCrossRefGoogle Scholar - 27.K. Nozari, P. Pedram, M. Molkara, Int. J. Theor. Phys.
**51**, 1268–1275 (2012)CrossRefGoogle Scholar - 28.A. Tawfik, A. Diab, Generalized uncertainty principle: approaches and applications. Int. J. Mod. Phys. D
**23.12**, 1430025 (2014)MathSciNetCrossRefGoogle Scholar - 29.H. Shababi, W.S. Chung, Phys. Lett. B
**770**, 445 (2017)ADSCrossRefGoogle Scholar - 30.K. Nouicer, Phys. Lett. B
**646**, 63 (2007)ADSMathSciNetCrossRefGoogle Scholar - 31.H. Shababi, P. Pedram, W.S. Chung, Int. J. Mod.Phys. A
**31**, 1650101 (2016)ADSCrossRefGoogle Scholar - 32.A. Ali, S. Das, E. Vagenas, Phys. Lett. B
**678**, 497 (2009)ADSMathSciNetCrossRefGoogle Scholar - 33.S. Das, E. Vagenas, A. Ali, Phys. Lett. B
**690**, 407 (2010); Erratum-ibid.**692**, 342 (2010)Google Scholar - 34.A. Ali, S. Das, E. Vagenas, Phys. Rev. D
**84**, 044013 (2011)ADSCrossRefGoogle Scholar - 35.Won Sang Chung, H. Hassanabadi, Phys. Lett. B
**785**, 127 (2018)ADSCrossRefGoogle Scholar - 36.J. Magueijo, L. Smolin, Phys. Rev. Lett.
**88**, 190403 (2002). arXiv:hep-th/0112090 ADSCrossRefGoogle Scholar - 37.J. Magueijo, L. Smolin, Phys. Rev. D
**71**, 026010 (2005). arXiv:hep-th/0401087 ADSMathSciNetCrossRefGoogle Scholar - 38.J.L. Cortes, J. Gamboa, Phys. Rev. D
**71**, 065015 (2005). arXiv:hep-th/0405285 ADSMathSciNetCrossRefGoogle Scholar

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