Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations

Chang, Lay Nam; Minić, Djordje; Okamura, N.; Takeuchi, Tatsu

doi:10.1103/physrevd.65.125027

Cited by 396 publications

(514 citation statements)

References 41 publications

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“…This assumption is not unjustifiable in view of the fact that the standard model charges cannot move in extra dimensions in theory of brane world model. Now, extra dimensions are widely accepted by theorists and there are large works about it [7][8][9][10][11][12][13]. This model gives us the new spectra, which looks same as our previous works [6], however, have different physical meaning.…”

Section: Introductionsupporting

confidence: 56%

Spectra of Hydrogen Atom with GUP and Extra Dimensions

Mu¹

2013

JMP

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We argue that the Generalized Uncertainty Ptinciple (GUP) and the compact Extra dimensions will lead the hydrogen atom has the non bounded state spectra which equal the free particle in extra dimensions. We use Generalized Uncertainty Principle (GUP) to calculate the new energy spectra of hydrogen atom, and which lives in three dimensional Euclidean spaces with one extra dimension. The result is not familiar with our known before that when n is large enough. In our modified new spectra, we obtain that, which is lager than zero.These new spectra give us new method to obtain the existence of extra dimensions. Finally, we find that the spectra are same as part of extra dimensions in planck scale.

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Section: Introductionsupporting

confidence: 56%

Spectra of Hydrogen Atom with GUP and Extra Dimensions

Mu¹

2013

JMP

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“…Among them, one may quote the momentum representation P = p, X = (1 + βp 2 )x (with x = id/dp) [9,10,19] and the (quasi)position representation X x, P p(1 + (1/3)βp 2 ) (with p = −id/dx) [12]. Whereas the former is exact, the latter is only valid to first order in β.…”

Section: Representations Of X and P In Terms Of Conventional Positionmentioning

confidence: 99%

“…When supplemented with shape invariance (SI) under parameter translation [25,26] or parameter scaling [27,28,29,30,31,32], these are known to provide a very powerful method for exactly solving problems in standard quantum mechanics. This is the case here too, since they have also allowed us [33] to deal with the one-dimensional harmonic oscillator in a uniform electric field and to algebraically rederive the results of [9,19].…”

Section: Introductionmentioning

confidence: 99%

“…To be able to solve the eigenvalue problem for such a Hamiltonian, we shall impose that (i) it is factorizable as shown in (2.14), where the operators B ± , now depending only on three real parameters g, s, r, are chosen in the form 19) and (ii) H is the first member H 0 of a hierarchy of Hamiltonians H i , i = 0, 1, 2, . .…”

Section: Applications To Quantum Mechanicsmentioning

confidence: 99%

“…Such problems include the one-and D-dimensional harmonic oscillators [9,19], for which exact solutions have been provided, the hydrogen atom, for which perturbative [12,14], numerical [13], or one-dimensional exact [20] results have been obtained, and the gravitational quantum well [15], which has been treated perturbatively. The use of WKB approximation has also been tested [21].…”

Section: Introductionmentioning

confidence: 99%

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Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics

Quesne¹

2007

SIGMA

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Abstract. Two generalizations of Kempf's quadratic canonical commutation relation in one dimension are considered. The first one is the most general quadratic commutation relation. The corresponding nonzero minimal uncertainties in position and momentum are determined and the effect on the energy spectrum and eigenfunctions of the harmonic oscillator in an electric field is studied. The second extension is a function-dependent generalization of the simplest quadratic commutation relation with only a nonzero minimal uncertainty in position. Such an uncertainty now becomes dependent on the average position. With each function-dependent commutation relation we associate a family of potentials whose spectrum can be exactly determined through supersymmetric quantum mechanical and shape invariance techniques. Some representations of the generalized Heisenberg algebras are proposed in terms of conventional position and momentum operators x, p. The resulting Hamiltonians contain a contribution proportional to p 4 and their p-dependent terms may also be functions of x. The theory is illustrated by considering Pöschl-Teller and Morse potentials.

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A minimal length uncertainty approach to cosmological constant problem

Diab

Tawfik

2021

Astron Nachr

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Based on quantum mechanical framework for the minimal length uncertainty, we demonstrate that the generalized uncertainty principle (GUP) parameter could be best constrained by recent gravitational waves observations on one hand. On the other hand, this suggests modified dispersion relations (MDRs) enabling an estimation for the difference between the group velocity of gravitons and that of photons. Utilizing features of the UV/IR correspondence and the obvious similarities between GUP (including non-gravitating and gravitating impacts on Heisenberg uncertainty principle) and the discrepancy between the theoretical and the observed cosmological constant (apparently manifesting gravitational influences on the vacuum energy density), we suggest a possible solution for the cosmological constant problem.

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Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations

Cited by 396 publications

References 41 publications

Spectra of Hydrogen Atom with GUP and Extra Dimensions

Spectra of Hydrogen Atom with GUP and Extra Dimensions

Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics

A minimal length uncertainty approach to cosmological constant problem

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