Schrödinger equation and resonant scattering in the presence of a minimal length

Haouat, S.

doi:10.1016/j.physletb.2013.12.060

Cited by 37 publications

(25 citation statements)

References 31 publications

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“…By following Ref. [38], we transformed this equation into a second order differential equation. We explicitly illustrated the regularizing effect that the fundamental length plays on the singularity of the problem.…”

Section: Discussionmentioning

confidence: 99%

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Singular inverse square potential in coordinate space with a minimal length

Bouaziz

Birkandan

2017

Annals of Physics

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The problem of a particle of mass m in the field of the inverse square potential α/r 2 is studied in quantum mechanics with a generalized uncertainty principle, characterized by the existence of a minimal length. Using the coordinate representation, for a specific form of the generalized uncertainty relation, we solve the deformed Schrödinger equation analytically in terms of confluent Heun functions. We explicitly show the regularizing effect of the minimal length on the singularity of the potential. We discuss the problem of bound states in detail and we derive an expression for the energy spectrum in a natural way from the square integrability condition; the results are in complete agreement with the literature.

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Section: Discussionmentioning

confidence: 99%

“…On another side, it has been recently proposed, in Ref. [38], an ad hoc transformation to reduce the order of the Schrödinger differential equation in coordinate space. Then, this approach has been applied to the spherical square well potential to investigate the consequence of the GUP on the resonant scattering.…”

Section: Introductionmentioning

confidence: 99%

Singular inverse square potential in coordinate space with a minimal length

Bouaziz

Birkandan

2017

Annals of Physics

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“…The deformed momentum operators will almost inevitably cause Hamiltonians of all quantum mechanical systems to be corrected. Since Kempf and his colleagues established the theoretical framework of quantum mechanics based on generalized uncertainty, the studies of Schrödinger equation, [13][14][15][16][17][18][19][20][21][22][23][24][25][26] the Dirac equation, [27][28][29][30][31][32][33][34][35][36][37][38] K-G equation [39][40][41] and DKP equation [42][43][44][45][46][47][48][49][50] get great interest, and some phenomena in black hole remnants, 51,52 the trans-Planckian problem of inflation, 53,54 and the cosmological constant problem 55,56 can be explained by generalized uncertainty relations. On the quantum level, except for the bound state, some related works on scattering state have been reported.…”

Section: Introductionmentioning

confidence: 99%

Scattering in graphene quantum dots under generalized uncertainty principle

Long

et al. 2019

Int. J. Mod. Phys. A

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In this article, the Dirac electron scattering problem on circular barrier of radius [Formula: see text] is studied under the generalized uncertainty principle (GUP). The expressions of scattering coefficients, scattering cross-section and scattering efficiency of massless Dirac particle are obtained by solving the massless Dirac equation under GUP and discussed by numerical methods. It shows that the scattering coefficient, the scattering cross-section, and the scattering efficiency depend explicitly on the GUP parameter [Formula: see text]. For the scattering coefficient [Formula: see text], GUP may cause slight shift in the oscillation position of [Formula: see text] and make some peaks value of [Formula: see text] smaller. For scattering cross-section and scattering efficiency, GUP may also lead to slight shift in their oscillation position and increase of amplitude when the GUP parameter increases.

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“…This explains why various physical problems are reconsidered by taking into account the minimal length. As example, we cite the harmonic oscillator [24][25][26], the Hydrogen atom [26][27][28][29][30][31][32], the inverse square potential [33], the Dirac oscillator [34], and the resonant scattering by a potential barrier [35,36]. Elsewhere, the influence of the minimal length on the Casimir effect has been communicated in several works [37,38].…”

Section: Introductionmentioning

confidence: 99%

Influence of a minimal length on the creation of scalar particles

Haouat

Nouicer

2014

Phys. Rev. D

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In this paper we have studied the problem of scalar particles pair creation by an electric field in the presence of a minimal length. Two sets of exact solutions for the Klein Gordon equation are given in momentum space. Then the canonical method based on Bogoliubov transformation connecting the "in" with the "out" states is applied to calculate the probability to create a pair of particles and the mean number of created particles. The number of created particles per unit of time per unit of length, which is related directly to the experimental measurements, is calculated. It is shown that, with an electric field less than the critical value, the minimal length minimizes the particle creation. It is shown, also, that the limit of zero minimal length reproduces the known results corresponding to the ordinary quantum fields.

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Schrödinger equation and resonant scattering in the presence of a minimal length

Cited by 37 publications

References 31 publications

Singular inverse square potential in coordinate space with a minimal length

Singular inverse square potential in coordinate space with a minimal length

Scattering in graphene quantum dots under generalized uncertainty principle

Influence of a minimal length on the creation of scalar particles

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