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Physics Essays

Classical physics derivation of quantization of electron elliptical orbit in hydrogenlike atom

Jiqing Zeng

Published Jun 10, 2022• 1 citation
Publisher

Abstract

On the basis of revising the quantum concept and reinterpreting Bohr's hydrogen atom structure model in classical physics, this paper deduces the elliptical orbital energy level formula of electrons in hydrogenlike atoms in detail. When the electron transition time is taken as unit time (1 s), the energy level formula is completely consistent with Bohr Sommerfeld's atomic structure theory.

DOI
10.4006/0836-1398-35.2.147

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On the basis of revising the quantum concept and reinterpreting Bohr's hydrogen atom structure model in classical physics, this paper deduces the elliptical orbital energy level formula of electrons in hydrogenlike atoms in detail. When the electron transition time is taken as unit time (1 s), the energy level formula is completely consistent with Bohr Sommerfeld's atomic structure theory.
This paper discusses Bohr-Somerfield quantization theory and tries to explain how all earlier propositions about energy at atomic level failed to correspond to observed facts and further how this model solved the mathematical discrepancies leading to a near perfect model of energy exchange at the atomic level. The author has attempted here to show the historical connections, logical justifications as well as mathematical expressions for the energy of electron in hydrogen atom and prove how Bohr’s celebrated theory of atomic structure is an application of Planck’s theory of quanta to the Rutherford nuclear atom in an attempt, extraordinary fruitful, to define the nature of the orbits in which the electrons might revolve round the nucleus and to explain the origin of spectral lines of the elements. It further shows how Sommerfeld extended the model so as to include elliptical orbits. The introduction of the elliptic orbits has no new energy for the electron of the hydrogen atom. Bohr had also solved the dilemma arose due to the Rutherford’s atomic model and he admitting the failure of the classical theory, applied with remarkable success of the quantum theory to the Rutherford nuclear atom with revolving electron, which leads us to understand the significance of the Bohr’s atom model.  Keywords: Electromagnetic theory, Model, Electrical attraction, Energy, Atom.
Niels Bohr’s model of the hydrogen and hydrogen-like atoms, in which the electron is allowed to revolve in certain circular orbits around the nucleus, was extended by Arnold Sommerfeld to allow for the possibility of elliptical orbits. Each elliptical orbit was constrained by two quantization conditions specified by a positive integer nφ and a non-negative integer nr. If, in the presence of the attractive Coulomb force between the electron and the nucleus, the dynamics of the atom were taken to be governed by the Newtonian laws of motion, the energy Ɛ of the two-particle system was found to be a function of the positive integer n = nr + nφ. The atomic energy levels thus exhibited an n-fold degeneracy, which would be lifted in the presence of external electric or magnetic fields, thus accounting for the fine-structure of the spectral lines emitted by the atom—phenomena observed, respectively, in the Stark effect and the Zeeman effect. Sommerfeld also examined the behavior of his elliptical orbits by allowing the revolving electron to obey Einstein’s relativistic dynamics. In this case, the energy-level degeneracies were lifted (even in the absence of external perturbations), and the elliptical orbits were found to precess around the nucleus—similarly to the precession of, say, the orbit of the planet Mercury around the Sun. What is astonishing about Sommerfeld’s relativistic expression of the energy Ɛ as a function of nφ and nr is that it is precisely the same expression as obtained for hydrogen and hydrogen-like atoms from Dirac’s equation, which came more than a decade later. Recall that the spin of the electron was not yet discovered when Sommerfeld developed his model, and that in Dirac’s fully-relativistic quantum mechanics, the spectral fine-structure arises from the coupling of the electron’s spin to its orbital motion. The present paper provides a streamlined yet detailed account of Sommerfeld’s elliptical orbits in hopes of bringing about a deeper appreciation for the miraculous coincidence of Sommerfeld’s result with that derived from Dirac’s equation.
Based on the basic coulomb field conservation situation,elliptic equation and quantization rules,a detail derivation process of computation of Sommerfeld's elliptical orbit of stationary state was given.Three equations of physical quantities and ellipse geometry parameters were concisely derived from three conservation situations with the aid of elliptic equation.Making up a new physical quantity,the relationship between the total energy and the equally kinetic of a cycle was given.Using two quantization rules on this base,a derivation on elliptic orbit and energy level of hydrogen-like atom was presented.