Relativistic extension of shape-invariant potentials

Alhaidari, A. D.

doi:10.1088/0305-4470/34/46/306

Cited by 75 publications

(79 citation statements)

References 16 publications

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“…Using this map and the known solutions (energy spectrum and wavefunctions) of the non-relativistic problem one can easily and directly obtain the relativistic spectrum and spinor wavefunctions. The main objective in all previous applications of the approach was in obtaining the relativistic energy spectrum and the spinor wavefunctions [1][2][3]6]. In this article, however, we demonstrate how one can utilize the same approach in generating the two-point Green's function − an important object of prime significance to the calculation of relativistic physical processes.…”

Section: Introductionmentioning

confidence: 90%

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The Dirac-Oscillator Green's Function

Alhaidari

2004

International Journal of Theoretical Physics

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We obtain the two-point Green's function for the relativistic DiracOscillator problem. This is accomplished by setting up the relativistic problem in such a way that makes comparison with the nonrelativistic problem highly transparent and results in a map of the latter into the former. The relativistic bound states energy spectrum is obtained by locating the energy poles of this Green's function in a simple and straightforward manner.

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Section: Introductionmentioning

confidence: 90%

“…It was applied successfully to the solution of various relativistic problems [1][2][3][4][5][6]. These included, but not limited to, the DiracCoulomb, Dirac-Morse, Dirac-Scarf, Dirac-Pöschl-Teller, Dirac-Woods-Saxon, etc.…”

Section: Introductionmentioning

confidence: 99%

The Dirac-Oscillator Green's Function

Alhaidari

2004

International Journal of Theoretical Physics

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“…Recently, many authors have worked on solving these equations with physical potentials including the Morse potential [3,4], Hulthén potential [5][6][7][8][9], Woods-Saxon potential [10,11], Pösch-Teller potential [12,13], reflectionless-type potential [14], pseudoharmonic oscillator [15][16][17], ring-shaped harmonic oscillator [18], V 0 tanh 2 ( / 0 ) potential [19], five-parameter exponential potential [20,21], Rosen-Morse potential [22], generalized symmetrical double-well potential [23], etc. It is remarkable that in most works in this area, the scalar and vector potentials are taken to be almost equal (i.e., S = V ) [2,24].…”

Section: Introductionmentioning

confidence: 99%

Relativistic solution in D-dimensions to a spin-zero particle for equal scalar and vector ring-shaped Kratzer potential

Ikhdair¹,

Sever

2008

Open Physics

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“…Eq. (10) (10) into the following one, which we choose to write in terms of the even potential component [2,3] …”

Section: Solving the Dirac Equation: In Atomic Units (M =mentioning

confidence: 99%

“…The main objective in all previous applications of the approach was in obtaining the discrete energy spectrum and spinor wavefunctions [2][3][4][5][6][7]. In this article, however, we demonstrate how one can utilize the same approach in generating the two-point Green's function which is an important object of prime significance in the calculation of physical processes where relativistic effects become relevant.…”

mentioning

confidence: 93%

The relativistic Dirac–Morse Green’s function

Alhaidari

2004

Journal of Mathematical Physics

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Using a recently developed approach for solving the three dimensional Dirac equation with spherical symmetry, we obtain the two-point Green's function of the relativistic Dirac-Morse problem. This is accomplished by setting up the relativistic problem in such a way that makes comparison with the nonrelativistic problem highly transparent and results in a mapping of the latter into the former. The relativistic bound states energy spectrum is easily obtained by locating the energy poles of the Green's function components in a simple and straightforward manner. Introduction: Despite all the work that has been done over the years on the Dirac equation, its exact solutions for local interaction has been limited to a very small set of potentials. Since the original work of Dirac in the early part of last century up until 1989 only the relativistic Coulomb problem was solved exactly. In 1989, the relativistic extension of the oscillator problem (Dirac-Oscillator) was finally formulated and solved by Moshinsky and Szczepaniak [1]. Recently, and in a series of articles [2-6], we presented an effective approach for solving the three dimensional Dirac equation for spherically symmetric potential interaction. The first step in the program started with the realization that the nonrelativistic Coulomb, Oscillator, and S-wave Morse problems belong to the same class of shape invariant potentials which carries a representation of so(2,1) Lie algebra. Therefore, the fact that the relativistic version of the first two problems (Dirac-Coulomb and Dirac-Oscillator) were solved exactly makes the solution of the third, in principle, feasible. Indeed, the relativistic Dirac-Morse problem was formulated and solved in Ref. 2. The bound state energy spectrum and spinor wavefunctions were obtained. Taking the nonrelativistic limit reproduces the familiar Schrödinger-Morse problem. Motivated by these findings, the same approach was applied successfully in obtaining solutions for the relativistic extension of yet another class of shape invariant potentials [3]. These included the Dirac-Scarf, Dirac-RosenMorse I & II, Dirac-Pöschl-Teller, and Dirac-Eckart potentials. Furthermore, using the same formalism quasi exactly solvable systems at rest mass energies were obtained for a large class of power-law relativistic potentials [4]. Quite recently, Guo Jian-You et al succeeded in constructing solutions for the relativistic Dirac-Woods-Saxon and DiracHulthén problems using the same approach [7]. In the fourth and last article of the series in our program of searching for exact solutions to the Dirac equation [5], we found a special graded extension of so(2,1) Lie algebra. Realization of this superalgebra by 2×2 matrices of differential operators acting in the two component spinor space was constructed. The linear span of this graded algebra gives the canonical form of the radial Dirac Hamiltonian. It turned out that the Dirac-Oscillator class, which also includes the Dirac-Coulomb and Dirac-Morse, carries a representation of this supersy...

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Relativistic extension of shape-invariant potentials

Cited by 75 publications

References 16 publications

The Dirac-Oscillator Green's Function

The Dirac-Oscillator Green's Function

Relativistic solution in D-dimensions to a spin-zero particle for equal scalar and vector ring-shaped Kratzer potential

The relativistic Dirac–Morse Green’s function

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