### Abstract

The effects of the generalized uncertainty principle (GUP) on the low-energy stationary states of a particle moving in a cavity with no sharp boundaries are determined by means of the perturbation expansion in the framework of onedimensional bandlimited quantum mechanics. A realization of GUP resulting in the existence of a finite ultraviolet (UV) wave-vector cutoff K ∼ 1/ℓP (with the Planck length ℓP) is considered. The cavity of the size ℓ 蠑 ℓP is represented by an infinitely deep trapezoid-well potential with boundaries smeared out in a range R satisfying the inequalities ℓ 蠑 R ≳ ℓP. In order to determine the energy shifts of the low-lying stationary states, the usual perturbation expansion is reformulated in a manner that enables one to treat consistently order-by-order the direct and indirect GUP effects, i.e., those due to the modification of the Hamiltonian and the lack of the UV modes, respectively. It is shown that the leading terms of the indirect and the direct GUP effects are of the first and second order, respectively, in the small parameter ℓP/ℓ in agreement with our previous finding in a more naive approach Sailer et al (2013 Phys. Rev. D 87 084056).

Original language | English |
---|---|

Article number | 075305 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 48 |

Issue number | 7 |

DOIs | |

Publication status | Published - Feb 20 2015 |

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### Keywords

- Bandlimited quantum mechanics
- Generalized uncertainty principle
- Minimal length uncertainty

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Modelling and Simulation
- Statistics and Probability

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*48*(7), [075305]. https://doi.org/10.1088/1751-8113/48/7/075305

**Particle in a cavity in one-dimensional bandlimited quantum mechanics.** / Sailer, K.; Péli, Z.; Nagy, S.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 48, no. 7, 075305. https://doi.org/10.1088/1751-8113/48/7/075305

}

TY - JOUR

T1 - Particle in a cavity in one-dimensional bandlimited quantum mechanics

AU - Sailer, K.

AU - Péli, Z.

AU - Nagy, S.

PY - 2015/2/20

Y1 - 2015/2/20

N2 - The effects of the generalized uncertainty principle (GUP) on the low-energy stationary states of a particle moving in a cavity with no sharp boundaries are determined by means of the perturbation expansion in the framework of onedimensional bandlimited quantum mechanics. A realization of GUP resulting in the existence of a finite ultraviolet (UV) wave-vector cutoff K ∼ 1/ℓP (with the Planck length ℓP) is considered. The cavity of the size ℓ 蠑 ℓP is represented by an infinitely deep trapezoid-well potential with boundaries smeared out in a range R satisfying the inequalities ℓ 蠑 R ≳ ℓP. In order to determine the energy shifts of the low-lying stationary states, the usual perturbation expansion is reformulated in a manner that enables one to treat consistently order-by-order the direct and indirect GUP effects, i.e., those due to the modification of the Hamiltonian and the lack of the UV modes, respectively. It is shown that the leading terms of the indirect and the direct GUP effects are of the first and second order, respectively, in the small parameter ℓP/ℓ in agreement with our previous finding in a more naive approach Sailer et al (2013 Phys. Rev. D 87 084056).

AB - The effects of the generalized uncertainty principle (GUP) on the low-energy stationary states of a particle moving in a cavity with no sharp boundaries are determined by means of the perturbation expansion in the framework of onedimensional bandlimited quantum mechanics. A realization of GUP resulting in the existence of a finite ultraviolet (UV) wave-vector cutoff K ∼ 1/ℓP (with the Planck length ℓP) is considered. The cavity of the size ℓ 蠑 ℓP is represented by an infinitely deep trapezoid-well potential with boundaries smeared out in a range R satisfying the inequalities ℓ 蠑 R ≳ ℓP. In order to determine the energy shifts of the low-lying stationary states, the usual perturbation expansion is reformulated in a manner that enables one to treat consistently order-by-order the direct and indirect GUP effects, i.e., those due to the modification of the Hamiltonian and the lack of the UV modes, respectively. It is shown that the leading terms of the indirect and the direct GUP effects are of the first and second order, respectively, in the small parameter ℓP/ℓ in agreement with our previous finding in a more naive approach Sailer et al (2013 Phys. Rev. D 87 084056).

KW - Bandlimited quantum mechanics

KW - Generalized uncertainty principle

KW - Minimal length uncertainty

UR - http://www.scopus.com/inward/record.url?scp=84921719470&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84921719470&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/48/7/075305

DO - 10.1088/1751-8113/48/7/075305

M3 - Article

AN - SCOPUS:84921719470

VL - 48

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 7

M1 - 075305

ER -