Relativistic Corrections to the Bohr Model of the Atom

Abstract

A simple means for extending the Bohr model of the atom to include relativistic corrections is presented. The derivation, which assumes circular orbits and a stationary nucleus, is similar to that for the non-relativistic case, except that the relativistic expressions for mass and kinetic energy are employed. Corrections, consistent with those of Sommerfeld, can thus be obtained to the radii and energy levels for circular orbits without recourse to the lengthy Sommerfeld procedure. The brevity of the derivation makes it suitable for inclusion in the standard first course in modern physics.

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

“…which is in complete agreement with relations (9) of Reference [4] and (6) of Reference [7]. The first term, E vib , is the nonrelativistic energy eigenvalue of HLAs and the second quadratic term (proportional to E 2 vib ) is concerned with the perturbation that occurs in HLAs due to the relativistic motion of the electron.…”

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

“…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…”

“…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.…”

“…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. For example, the relativistic angular frequency ω rel n is directly increased by γ, while the relativistic radius r rel n is reduced by the inverse Lorentz factor γ À1 , so that the rotational speed of electron remains unchanged in the Bohr circular orbits [4,7]. The energy eigenvalue and Hamiltonian are simultaneously modified by some relativistic perturbation terms as mentioned in many studies [8,11,12].…”