Evaluation of matrix elements ofrkand recurrence relations for the Klein–Gordon equation with a Coulomb field

Chen, Changyuan; Dong, Shi‐Hai

doi:10.1088/0031-8949/73/5/016

Cited by 4 publications

(6 citation statements)

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“…( 9) is always (i.e. for any observer's velocity v) appropriately normalized while the density ρ N LI ( r) = |ψ nlm ( r)| 2 (used in [17]) is not. This is numerically illustrated in Figure 1 for a pionic atom with nuclear charge Z = 68 in the infinite nuclear mass approximation (π − -meson mass=273.132054 a.u.).…”

Section: Introductionmentioning

confidence: 99%

Relativistic Klein–Gordon charge effects by information-theoretic measures

2010

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The charge spreading of ground and excited states of Klein-Gordon particles moving in a Coulomb potential is quantitatively analyzed by means of the ordinary moments and the Heisenberg measure as well as by use of the most relevant information-theoretic measures of global (Shannon entropic power) and local (Fisher's information) types. The dependence of these complementary quantities on the nuclear charge Z and the quantum numbers characterizing the physical states is carefully discussed. The comparison of the relativistic Klein-Gordon and nonrelativistic Schrödinger values is made. The non-relativistic limits at large principal quantum number n and for small values of Z are also reached.

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Section: Introductionmentioning

confidence: 99%

Relativistic Klein–Gordon charge effects by information-theoretic measures

2010

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“…muonic and pionic atoms [23]). Only recently, Chen and Dong [22] have been able to calculate explicit expressions for these moments and some off-diagonal matrix elements of r k for a Klein-Gordon single-particle of mass m 0 in the Coulomb potential V ( r) = − Ze 2 r . These authors, however, do not use the Lorentz-Invariant (LI) Klein-Gordon charge density…”

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confidence: 99%

“…Let us emphasize that the resulting Lorentz-invariant charge density ρ LI ( r) given by Eqs. ( 3)-( 10) is always (i.e., for any observer's velocity v) appropriately normalized, while the non-Lorentz-invariant density ρ N LI ( r) used by Cheng and Dong [22] is not. This was numerically discussed in Ref [24] for some pionic atoms in the infinite nuclear mass approximation.…”

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“…but just the Non-Lorentz-Invariant (NLI) expression ρ N LI ( r) = |Ψ nlm ( r)| 2 as in the non-relativistic case, where ǫ and Ψ( r) denote the physical solutions of the Klein-Gordon equation[19,22] …”

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confidence: 99%

“…The Klein-Gordon wave equation was introduced in 1926 and constituted the first theoretical description of particle dynamics in a relativistic quantum setting [18]. Since then, the study of its properties for different potentials of various dimensionalities has been a problem of increasing interest [19][20][21][22]. Many efforts were addressed to obtain the spectrum of energy levels and the ordinary moments or expectation values r k of the charge distribution of numerous single-particle systems (such as, e.g., muonic and pionic atoms [23]).…”

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Complexity analysis of Klein-Gordon single-particle systems

Manzano¹,

López-Rosa²,

Dehesa³

2010

EPL

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The Fisher-Shannon complexity is used to quantitatively estimate the contribution of relativistic effects to on the internal disorder of Klein-Gordon single-particle Coulomb systems which is manifest in the rich variety of three-dimensional geometries of its corresponding quantum-mechanical probability density. It is observed that, contrary to the non-relativistic case, the Fisher-Shannon complexity of these relativistic systems does depend on the potential strength (nuclear charge). This is numerically illustrated for pionic atoms. Moreover, its variation with the quantum numbers (n, l, m) is analysed in various ground and excited states. It is found that the relativistic effects enhance when n and/or l are decreasing. PACS numbers: 89.70.Cf, 03.65.Pm Numerous phenomena and properties of manyelectron systems have been qualitatively characterized by information-theoretic means. In particular, various single and composite information-theoretic measures have been proposed to identify and analyze the multiple facets of the internal disorder of non-relativistic quantum systems; see e.g. Ref. [1][2][3][4][5][6][7][8][9].They are often expressed as products of two quantities of local (e. g. the Fisher information) and/or global (e. g. the variance or Heisenberg measure, the Shannon entropy, the Renyi and Tsallis entropies and the disequilibrium or linear entropy ρ ) character, which describe the charge spreading of the system in a complementary and more complete manner than their individual components. This is the case of the disequilibrium-Shannon or Lopez-Ruiz-Mancini-Calvet (LMC in short) [2], disequilibrium-Heisenberg [4], Fisher-Shannon [5,8] and the Cramer-Rao [5, 10] complexities, which have their minimal values at the extreme ordered and disordered limits.Recently these studies have been extended to take into account the relativistic effects in atomic physics. Relativistic quantum mechanics [11] tells us that special relativity provokes (at times, severe) spatial modifications of the electron density of many-electron systems, what produces fundamental and measurable changes in their physical properties. The qualitative and quantitative evaluation of the relativistic modification of the spatial redistribution of the electron density of ground and excited states in atomic and molecular systems by informationtheoretic means is a widely open field. In the last three years the relativistic effects of various single and composite information-theoretic quantities of the ground states of hydrogenic [12] and neutral atoms [13][14][15] have been investigated in different relativistic settings.First Borgoo et al [13] (see also [16]), in a Dirac-Fock setting, find that the LMC shape complexity of the * Electronic address: manzano@ugr.es † Electronic address: slopez@ugr.es ‡ Electronic address: dehesa@ugr.es ground-state atoms (i) has an increasing dependence on the nuclear charge (also observed by Katriel and Sen [12] in Dirac ground-state hydrogenic systems), (ii) manifest shell and relativistic effects, the latter b...

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Analysis of N-dimensional Klein–Gordon equation for hydrogen molecule in the non-central potential field

Özfidan

Durmus

2018

Annals of Physics

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Evaluation of matrix elements ofr^kand recurrence relations for the Klein–Gordon equation with a Coulomb field

Cited by 4 publications

References 25 publications

Relativistic Klein–Gordon charge effects by information-theoretic measures

Relativistic Klein–Gordon charge effects by information-theoretic measures

Complexity analysis of Klein-Gordon single-particle systems

Analysis of N-dimensional Klein–Gordon equation for hydrogen molecule in the non-central potential field

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