Minimal length uncertainty relation and the hydrogen atom

Abstract

Abstract. We propose a new approach to calculate perturbatively the effects of a particular deformed Heisenberg algebra on an energy spectrum. We use this method to calculate the harmonic oscillator spectrum and find that the corrections are in agreement with a previous calculation. Then, we apply this approach to obtain the hydrogen atom spectrum and we find that splittings of degenerate energy levels appear. Comparison with experimental data yields an interesting upper bound for the deformation parameter of the Heisenberg algebra.

Generalized uncertainty principle in Bianchi type I quantum cosmology

“…More general cases of such commutation relations are studied in [7]- [11]. Also various applications of the low energy effects of the modified Heisenberg uncertainty relations have been extensively studied, see for example [12]- [15]. In this letter we consider a two dimensional minisuperspace of Bianchi type I cosmology in the GUP framework.…”

No existence of black holes at LHC due to minimal length in quantum gravity

“…From Black hole physics, the Uncertainty Principle, ∆x ∼ /∆p, is modified at the Planck energy scale, when the corresponding Schwarzschild radius is comparable to the Compton wavelength (both are approximately equal to the Planck length). Higher energies result in a further increase of the Schwarzschild radius, resulting in ∆x ≈ ℓ 2 P l ∆p/ The above observation, along with a combination of thought experiments and rigorous derivations suggest that the Generalized Uncertainty Principle (GUP) holds at all scales, and is represented by [3][4][5][6][7][8] ∆x i ∆p i ≥ 2 [1 + β (∆p) 2 + < p > 2 It was shown in [5], that inequality (1.1) is equivalent to the following modified Heisenberg algebra [x i , p j ] = i (δ ij + βδ ij p 2 + 2βp i p j ) . Recently, we proposed the GUP in [10][11][12] which predicts maximum observable momenta besides the existence of minimal measurable length and is consistent with Doubly Special Relativity (DSR) theories, String Theory and Black Holes Physics and which ensures [x i , x j ] = 0 = [p i , p j ] (via the Jacobi identity).…”

“…In this section, we make analysis of BH thermodynamics If GUP proposed in [3][4][5][6][7] is taken into consideration.…”

“…In Sec .3, we investigate BH thermodynamics if GUP of Eq. (1.1) which was proposed earlier by [3][4][5][6][7] is taken into consideration. We found that the BH minimal mass could be detected in the range between 60 TeV and 2.3×10 5 TeV depending on the number of extra dimensions for D = 6..10.…”

Quantum tunneling from a high dimensional Gödel black hole

“…We can refer to [47][48][49], setting the time t as t → t − πi/2κ an additional contribution coming from the time part of the action can be discoved that is Im[(ω − JΩ + )∆t out, in ] = −π(ω − JΩ + )/2κ. Eventually, incorporating the temporal contribution, we turns out to be of the form…”