Exact solutions of the Dirac equation for Makarov potential by means of the quantum Hamilton–Jacobi formalism

Touloum, S.; Gharbi, Abdelhakim; Bouda, A.

doi:10.1007/s12648-016-0941-7

Cited by 6 publications

(4 citation statements)

References 19 publications

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“…The solutions of the spin and pseudospin symmetries of the Dirac equation for various noncentral potentials have been derived. For exactly solvable potentials, we can cite the Hartmann potential [25][26][27][28][29], the ring-shaped non-spherical harmonic oscillator [30,31], the Makarov potential [32], the new ring-shaped non-spherical harmonic oscillator [33][34][35], the pseudo-harmonic oscillatory ring-shaped potential [36,37] and others [38][39][40][41][42][43][44][45][46][47][48][49][50][51]. However, for the potentials such as the ring-shaped generalized Hulthén potential [52], the Manning-Rosen potential plus a ring-shaped like potential [53], the ring-shaped Woods-Saxon potential [54], the ring-shaped q − deformed Woods-Saxon potential [55], the q − deformed hyperbolic Pöschl-Teller potential and trigonometric Scarf II noncentral potential [56], The Eckart potential and trigonometric Manning-Rosen potential [57] and the modified Pöschl-Teller potential and trigonometric Scarf II noncentral potential [58], the Dirac equation has been solved by using an appropriate approximation of the centrifugal term.…”

Section: Introductionmentioning

confidence: 99%

Complete solutions of the Dirac equation with q-deformed hyperbolic Pöschl-Teller potential plus a trigonometric Scarf II potential

Kadja

2024

Phys. Scr.

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Under the condition of the spin symmetry, we rigorously solve the Dirac equation with the q-deformed hyperbolic Pöschl-Teller potential plus a trigonometric Scarf II potential. According to the values of the deformation parameter q, four different cases are considered. For the two cases q≥1 and q≤-1 with (1/(2α))ln|q|<r<+∞, the analytical energy spectra and the spinor wave functions associated with the l_{d}-wave bound states are obtained using a suitable approximation to the centrifugal potential term. When 0<q<1 or -1<q<0, the spinor wave functions for the s-wave bound states are derived and we find that the quantization conditions are transcendental equations which can be solved numerically. The special case q=0 is also discussed.

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Section: Introductionmentioning

confidence: 99%

Complete solutions of the Dirac equation with q-deformed hyperbolic Pöschl-Teller potential plus a trigonometric Scarf II potential

Kadja

2024

Phys. Scr.

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“…To this end and over the years, numerous resolution methods have been developed. We can cite the Nikiforov-Uvarov method [1,2], the factorization method [3], the path integral formalism [4], supersymmetric quantum mechanics [5], the integral equation method [6], the group-theoretical method [7], the quantum Hamilton-Jacobi formalism [8][9][10][11][12][13], the Ma and Xu's exact quantization rule [14][15][16][17]. the latter, which can be viewed as a generalization of the famous WKB quantization method, belongs to the family of quantization rules that are well known to be efficient to obtain, exactly or approximatively, energy spectra without solving the Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

Bound states of Dirac equation using the proper quantization rule

Bachi¹,

Touloum²,

Ighezou³

et al. 2021

Phys. Scr.

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Using the proper quantization rule, we investigate the exact solution of Dirac equation for Hartmann and the ring-shaped non-spherical harmonic oscillator potentials under the condition of equal scalar and vector potentials. By considering the proper quantization condition within angular and radial variables, the exact relativistic energy spectra are obtained for each system. Then by the mean of suitable changes of variables, the corresponding spinor wave-functions are constructed where the normalization constants are exactly calculated. We also derived the non-relativistic limit of energy spectra.

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“…Moreover, the discrete fractional solutions of radial Schrödinger equation for Makarov potential have been studied . Other studies related to this potential are also carried out . It should be recognized that the main contributions to this topic have been made in the nonrelativistic Schrödinger equation case.…”

Section: Introductionmentioning

confidence: 99%

“…[15] Other studies related to this potential are also carried out. [16][17][18][19][20][21] It should be recognized that the main contributions to this topic have been made in the nonrelativistic Schrödinger equation case. However, the degenerate states for the nonrelativistic systems with the same energy levels cannot be distinguished.…”

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confidence: 99%

Study of spin‐orbit interaction for the Makarov potential

Chen

You

et al. 2018

Int J of Quantum Chemistry

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We study the spin‐orbit interaction (SOI) for the Makarov potential to correct the nonrelativistic Schrödinger energy levels and discuss the degeneracy of the energy levels. For a certain principal quantum number n, when we do not consider the spin the degeneracy of the energy levels with magnetic quantum number m = 0 is n, while the degeneracy with the magnetic quantum number m ≠ 0 becomes 2(n − |m|). However, when we consider the spin s = 1/2 the degeneracy of the energy levels with magnetic quantum number m = 0 is 2n, while the degeneracy with the magnetic quantum number m ≠ 0 is 4(n − |m|). After taking SOI into account, the degeneracy in this case is still 2n since the energy levels with the magnetic quantum number m = 0 are not split, but the energy levels with the magnetic quantum number m ≠ 0 will be split into 2(n − |m|) energy levels, each of which has the degeneracy 2. The size and sequence of the energy level splitting are relevant for the potential parameters β and γ except for depending on those quantum numbers n, l, and m. We find that the degenerate energy levels are reversed at the critical values by considering the SOI effect.

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Exact solutions of the Dirac equation for Makarov potential by means of the quantum Hamilton–Jacobi formalism

Cited by 6 publications

References 19 publications

Complete solutions of the Dirac equation with q-deformed hyperbolic Pöschl-Teller potential plus a trigonometric Scarf II potential

Complete solutions of the Dirac equation with q-deformed hyperbolic Pöschl-Teller potential plus a trigonometric Scarf II potential

Bound states of Dirac equation using the proper quantization rule

Study of spin‐orbit interaction for the Makarov potential

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