A simple relativistic Bohr atom

Terzis, Andreas

doi:10.1088/0143-0807/29/4/008

Cited by 9 publications

(18 citation statements)

References 23 publications

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“…It is evident from () that the potential energy increases with the Lorentz factor

γ

from the relativistic point of view. One can now determine the relativistic energy eigenvalue (relativistic vibrational energy) by summing the relativistic kinetic and potential energies () and () as

E_{italicvib}^{italicrel} = - \frac{1}{2} m_{e} c^{2} \frac{{(italiczα)}^{2}}{n^{2}} - \frac{1}{8} m_{e} c^{2} \frac{{(italiczα)}^{4}}{n^{4}} = E_{italicvib} - \frac{1}{2 m_{e} c^{2}} E_{italicvib}^{2},

which is in complete agreement with relations (9) of Reference [4] and (6) of Reference [7]. The first term,

E_{italicvib}

, is the nonrelativistic energy eigenvalue of HLAs and the second quadratic term (proportional to

E_{italicvib}^{2}

) is concerned with the perturbation that occurs in HLAs due to the relativistic motion of the electron.…”

Section: Relativistic Kinetic Potential and Vibrational Energiessupporting

confidence: 78%

“…(5) of References [4,7], respectively, but disagrees with the corresponding relation (13) of Reference [8] in which the contraction factor is consid-…”

Section: Relativistic Kinetic Potential and Vibrational Energiesmentioning

confidence: 66%

“…where m e is the rest mass of the electron and p rel ¼ γ m e v n is the relativistic linear momentum of the electron in Bohr circular orbits with constant speed v n ¼ ze 2 =nℏ ¼ zαc=n (1=4πε 0 ¼ 1) and the Lorentz factor γ ¼ 1 À z 2 α 2 =n 2 À Á À1=2 [4,7,12]. As a result, the relativistic kinetic energy (1) can be approximated up to the fourth power zα ð Þ 4 =n 4 after substituting the expansions of γ 2 and γ 4 as…”

Section: Relativistic Kinetic Potential and Vibrational Energiesmentioning

confidence: 99%

“…It is worth noting that these principles play a vital role on a smaller scale in the modification of the natural characteristics of lighter elements. These characteristics include energy eigenvalue, Hamiltonian, angular frequency and radius of a single oscillating electron as well as entropy of hydrogen-like atoms (HLAs) [4][5][6][7][8][9][10]. Here, we demonstrate that the Lorentz factor γ ¼ 1 À z 2 α 2 =n 2 À Á À1=2 (with z, α, and n the atomic number, fine structure constant, and energy level, respectively) is responsible for the relativistic corrections in physical and chemical quantities.…”

Section: Introductionmentioning

confidence: 99%

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Relativistic ro‐vibrational feature of electron in Bohr's orbits of hydrogen‐like atoms in Heisenberg picture

Jahanpanah

Vahedi

Khosrojerdi

2022

Int J of Quantum Chemistry

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The relativistic effects have been widely reported to affect the chemical and physical properties of the heavy elements such as lanthanides (57≤z≤71), actinides (89≤z≤103), and transactinides (z≥104). This effect is definitely weakened by reducing the atomic number z in lighter elements such as hydrogen‐like atoms (HLAs). The aim of present paper is to investigate the relativistic effects of electron motion in Bohr orbits on the chemical and physical properties of HLAs. The theoretical model is based on the Heisenberg (rather than Schrodinger) picture where the relativistic vibrational Hamiltonian (RVH) Hitalicvibitalicrel is expanded as a power series of harmonic oscillator Hamiltonian H0 for the first time. By applying the first‐order RVH (correct to H0) to the Heisenberg equation, a pair of coupled equations is obtained for the relativistic position and linear momentum of electron. A simple comparison of the first‐order relativistic and nonrelativistic equations reveals that the relativistic natural frequency of an HLA (like entropy) is slowly raised by increasing z beyond z≈20. In general, RVH plays a fundamental role because it specifies the temporal relativistic variations of position, velocity, and linear momentum of the oscillating electron. The results are finally verified by demonstrating energy conservation.

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“…It is evident from () that the potential energy increases with the Lorentz factor

γ

from the relativistic point of view. One can now determine the relativistic energy eigenvalue (relativistic vibrational energy) by summing the relativistic kinetic and potential energies () and () as

E_{italicvib}^{italicrel} = - \frac{1}{2} m_{e} c^{2} \frac{{(italiczα)}^{2}}{n^{2}} - \frac{1}{8} m_{e} c^{2} \frac{{(italiczα)}^{4}}{n^{4}} = E_{italicvib} - \frac{1}{2 m_{e} c^{2}} E_{italicvib}^{2},

which is in complete agreement with relations (9) of Reference [4] and (6) of Reference [7]. The first term,

E_{italicvib}

, is the nonrelativistic energy eigenvalue of HLAs and the second quadratic term (proportional to

E_{italicvib}^{2}

) is concerned with the perturbation that occurs in HLAs due to the relativistic motion of the electron.…”

Section: Relativistic Kinetic Potential and Vibrational Energiessupporting

confidence: 78%

“…(5) of References [4,7], respectively, but disagrees with the corresponding relation (13) of Reference [8] in which the contraction factor is consid-…”

Section: Relativistic Kinetic Potential and Vibrational Energiesmentioning

confidence: 66%

Section: Relativistic Kinetic Potential and Vibrational Energiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Relativistic ro‐vibrational feature of electron in Bohr's orbits of hydrogen‐like atoms in Heisenberg picture

Jahanpanah

Vahedi

Khosrojerdi

2022

Int J of Quantum Chemistry

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“…In the development of relativistic approach, Terzis et al expressed relativistic version of the Bohr radii for hydrogen-like atoms with circular orbits as [22],…”

Section: An Extension Of Kinetic Energy Uncertainty Analysis For Relativistic Bohr Hydrogen-like Atomsmentioning

confidence: 99%

Lateral Position Uncertainty of Electrons in Bohr Hydrogen-like Atoms: An Implication of Heisenberg Uncertainty Principle

Alagoz¹

2019

Adıyaman University Journal of Science

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This paper presents a theoretical investigation on effects of lateral position uncertainty of captivity electrons within spherical electron shells of Bohr hydrogen-like atoms. A captivity electron, which is spatially confined in Bohr orbits, introduces a lateral position uncertainty that can be determined by considering the area of the electron shell. After deriving uncertainty relation for position and kinetic energy, author theoretically demonstrates that, due to the lateral position uncertainties of electrons in spherical shells, Heisenberg uncertainty principle suggests uncertainty bounds in measurement of kinetic energy states of captivity electrons that orbits non-relativistic hydrogen-like Bohr atom. Afterward, these analyses are extended for relativistic hydrogen-like Bohr atom case.

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On the chemical bond complexity of the H2+ in 1-D: The ground-state avoided crossing

López‐Castillo

2021

Computational and Theoretical Chemistry

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A simple relativistic Bohr atom

Cited by 9 publications

References 23 publications

Relativistic ro‐vibrational feature of electron in Bohr's orbits of hydrogen‐like atoms in Heisenberg picture

Relativistic ro‐vibrational feature of electron in Bohr's orbits of hydrogen‐like atoms in Heisenberg picture

Lateral Position Uncertainty of Electrons in Bohr Hydrogen-like Atoms: An Implication of Heisenberg Uncertainty Principle

On the chemical bond complexity of the H2+ in 1-D: The ground-state avoided crossing

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