A relativistic model of the isotropic three-dimensional singular oscillator

Nagiyev, Sh. M.; Jafarov, E. I.; Imanov, R. M.; Homorodean, Laurean

doi:10.1016/j.physleta.2004.11.021

Cited by 11 publications

(14 citation statements)

References 28 publications

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“…As this Appendix was approaching completion, the highly relevant article of Nagiyev et al [30] appeared. Their work proposes a relativistic model of the isotropic three-dimensional singular oscillator.…”

Section: Appendix 1 Relativistic Effects For the Chemical Potential mentioning

confidence: 99%

“…(A15) has not been derived. Nagiyev et al [30] set up a relativistic model of the isotropic three-dimensional singular oscillator in terms of a difference equation for the radial wave functions [their Eq. (24)].…”

Section: Appendix 1 Relativistic Effects For the Chemical Potential mentioning

confidence: 99%

“…(24)]. One of the achievements of their work is to obtain these relativistic wave functions in terms of the so-called continuous dual Hahn polynomials [30]. Nagiyev et al then show that in the nonrelativistic limit in which the Compton wavelength tends to zero, they recover exactly the nonrelativistic Schrö dinger wave functions and energy levels of this singular oscillator.…”

Section: Appendix 1 Relativistic Effects For the Chemical Potential mentioning

confidence: 99%

“…Finally, some proposals are put forward for future work, both theoretical and experimental, bearing on relativistic DFT formulated in terms of low-order difference equations. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem 105: 701-708, 2005 Key words: relativistic electron density; difference equations 1. Background and Outline S ome two decades ago, in an article on density functional theory (DFT) and its applications, Callaway and the author [1] drew attention to the fact that, whereas real atoms have ground states, relativistic Hamiltonians do not.…”

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Relativistic density functional theory posed in terms of difference equations

March

2005

Int J of Quantum Chemistry

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ABSTRACT:Because real atoms have ground states, while relativistic Hamiltonians do not, there is a first-principles reason to treat relativistic problems by means of density functional theory (DFT). Here, such a relativistic DFT is presented in terms of difference equations, which arise in turn from discretized differential equations. Two explicit examples considered are: (i) harmonically confined independent Fermions filling an arbitrary number of closed shells, and (ii) hydrogen-like atomic ions for arbitrary atomic number Z. These findings are then compared and contrasted with wave function theories, going back at least to Wall. Finally, some proposals are put forward for future work, both theoretical and experimental, bearing on relativistic DFT formulated in terms of low-order difference equations. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem 105: 701-708, 2005 Key words: relativistic electron density; difference equations 1. Background and Outline S ome two decades ago, in an article on density functional theory (DFT) and its applications, Callaway and the author [1] drew attention to the fact that, whereas real atoms have ground states, relativistic Hamiltonians do not. Even at that early stage, it was tempting to build a relativistic DFT without appeal to such Hamiltonians. Subsequent work by Holas and March [2,3] therefore attempted to construct a relativistic theory directly in terms of the so-called density amplitude (r) 1/2 , where (r), as usual, is the ground-state electron density.Progress in nonrelativistic DFT, going back to the work of Lawes and March [4], is possible, at least for some currently important problems such as harmonic confinement, in terms of a linear differential equation to solve for the density (r) itself, rather than the nonlinear equation for the density amplitude referred to above. Specifically, for harmonically confined independent Fermions filling an arbitrary number of closed shells, the nonrelativistic ground-state Fermion density (r) satisfies in ddimensions the linear third-order homogeneous differential equation [5]:Here is the characteristic frequency associated with the harmonic confinement, and (M ϩ 1) de- We will return briefly to the nonrelativistic solution of Eq. (1) in three dimensions below. At this point, we record unorthodox developments in relativistic wave function theory, which we compare and contrast with the relativistic DFT approach, the focus of the present study (discussed in detail later in this work). These "unorthodox" wave function treatments go back, at least, to Wall [6], with related and independent work more than a decade later by Ruijsenaars [7] and by Ord and Mann [8]. Some details are recorded below, especially in Section 4.3 and Appendix A2. Suffice it to say here that Wall [6] took the Schrö dinger differential equation for free electrons and converted it into a "relativistic wave function difference equation" by introducing a finite interval, the scale of which was determined by the Compton wavelength ϭ h/m 0 c, where m 0 now denote...

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Section: Appendix 1 Relativistic Effects For the Chemical Potential mentioning

confidence: 99%

Section: Appendix 1 Relativistic Effects For the Chemical Potential mentioning

confidence: 99%

Section: Appendix 1 Relativistic Effects For the Chemical Potential mentioning

confidence: 99%

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confidence: 99%

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Relativistic density functional theory posed in terms of difference equations

March

2005

Int J of Quantum Chemistry

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“…Later, this model was generalized for 3D case in [12]. Our model was formulated in the framework of the finite-difference version of the relativistic quantum mechanics, developed in [13][14][15][16][17].…”

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On a dynamical symmetry group of the relativistic linear singular oscillator

2006

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Abstract. -An exact approach for the factorization of the relativistic linear singular oscillator is proposed. This model is expressed by the finite-difference Schrödinger-like equation. We have found finite-difference raising and lowering operators, which are with the Hamiltonian operator form the close Lie algebra of the SU (1, 1) group.Introduction. -The singular harmonic oscillator is one of the rare exactly solvable problems in non-relativistic quantum mechanics [1,2]. This model is useful to explain many phenomena, such as description of interacting many-body systems [3], diatomic [4] and polyatomic [5] molecules, spin chains [6], quantum Hall effect [7], fractional statistics and anyons [8]. In spite of many interesting papers devoted to the study of the non-relativistic singular harmonic oscillator model [9], the number of works studying relativistic approachs to singular oscillator exact solution is still rather few [10].Recently, we constructed the exactly solvable relativistic model of the quantum linear singular oscillator [11]. Later, this model was generalized for 3D case in [12]. Our model was formulated in the framework of the finite-difference version of the relativistic quantum mechanics, developed in [13][14][15][16][17]. As we noted in [11], unlike the case of the Coulomb potential, the relativistic generalization of the oscillator or singular oscillator potentials are not uniquely defined. Therefore, one of the main requirements for proposed relativistic models is existence of their dynamical symmetry. In this Letter, we present the simplest way for construction of the close Lie algebra of the SU (1, 1) group for a relativistic model of the linear singular oscillator. In spite of expression of the problem under consideration by the finite-difference equation, it is also exactly factorizable and is in complete analogy with relevant non-relativistic problem. This factorization will allow us to obtain the exact finite-difference expression of the generators of SU (1, 1) group.

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Solution of the Logunov–Tavkhelidze Equation for the Three-Dimensional Oscillator Potential in the Relativistic Configuration Representation

Grishechkin

Kapshaǐ

2018

Russ Phys J

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A relativistic model of the isotropic three-dimensional singular oscillator

Cited by 11 publications

References 28 publications

Relativistic density functional theory posed in terms of difference equations

Relativistic density functional theory posed in terms of difference equations

On a dynamical symmetry group of the relativistic linear singular oscillator

Solution of the Logunov–Tavkhelidze Equation for the Three-Dimensional Oscillator Potential in the Relativistic Configuration Representation

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