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arXiv:1302.6163

Quantized magnetic flux through the orbits of hydrogen-like atoms within the atomic model of Sommerfeld

W.‐D. Stein

Published Feb 25, 2013• 1 citation
arXivHTMLPDF

Abstract

Within the Sommerfeld atomic model the quantization of magnetic flux through the electronic orbits is investigated together with its dependency on additional sources of magnetic fields. These sources alter the magnetic flux through the atomic orbits and in consequence are causing energy shifts. This effect is investigated for the cases, where the source is an external magnetic field, the magnetic moment of the nucleus or the magnetic moment of the electron. The energy shifts due to external magnetic fields, the magnetic dipole contribution of the hyperfine splitting and the spin-orbit coupling can be reproduced very well. The meaning of 'spin', however, changes within this approach drastically. The unusual Landé g-factor of 2 for the electron is a result of the orbital motion and the magnetic moment of the electron rather than it is an intrinsic property of the electron.

DOI
10.48550/arxiv.1302.6163

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Within the Sommerfeld atomic model the quantization of magnetic flux through the electronic orbits is investigated together with its dependency on additional sources of magnetic fields. These sources alter the magnetic flux through the atomic orbits and in consequence are causing energy shifts. This effect is investigated for the cases, where the source is an external magnetic field, the magnetic moment of the nucleus or the magnetic moment of the electron. The energy shifts due to external magnetic fields, the magnetic dipole contribution of the hyperfine splitting and the spin-orbit coupling can be reproduced very well. The meaning of 'spin', however, changes within this approach drastically. The unusual Landé g-factor of 2 for the electron is a result of the orbital motion and the magnetic moment of the electron rather than it is an intrinsic property of the electron.
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.
We consider a semi-classical model to describe the origin of the spin–orbit interaction in a simple system such as the hydrogen atom. The interaction energy is calculated in the rest-frame of the nucleus, around which an electron, having linear velocity and magnetic dipole-moment travels in a circular orbit. The interaction energy is due to the coupling of the induced electric dipole with the electric field of the nucleus. Assuming the radius of the electron's orbit remains constant during a spin-flip transition, our model predicts that the energy of the system changes by the factor emerging naturally as a consequence of equilibrium and the change of the kinetic energy of the electron. The correct factor for the spin–orbit coupling energy is thus derived without the need to invoke the well-known Thomas precession in the rest-frame of the electron.
This chapter discusses the prime cause of the magnetic moment within an individual atom. In the classical model, the angular momentum of the electrons can be used to determine the magnetic moments of the electrons by invoking the concept of electrical charge in motion. There are two contributions to the electronic magnetic moment: an orbital magnetic moment due to orbital angular momentum and a spin magnetic moment due to electron spin. The component of orbital angular momentum along the axis of a magnetic field is restricted to discrete values by quantum mechanics. Electrons have spin angular momentum, which can be represented by the spin quantum number. As the magnetic field is increased and the field splitting becomes greater than the multiple splitting, the anomalous Zeeman effect changes over to a normal Zeeman effect. The chapter looks at experimental evidence confirming the quantization of magnetic moments, specifically the Zeeman effect and the Stern-Gerlach experiment.
We have calculated the changes in the quantized magnetic fluxes corresponding to the electronic orbits of Dirac hydrogen atom within the framework of Dirac theory and found that the flux change between the final state and the initial state is equal to (±Φ 0 or zero).We have also calculated the non-zero matrix elements for dipole transitions (such as 1s↔2p and 2p↔3d) for the solutions of Dirac hydrogen atom. We find a complete agreement among all the above results. The combined result is that photon should carry an intrinsic flux quantum of (Φ 0 =hc/e).
We consider a semi-classical model to describe the origin of the spin-orbit interaction in a simple system such as the hydrogen atom. The interaction energy U is calculated in the rest-frame of the nucleus, around which an electron, having linear velocity v and magnetic dipole-moment μ, travels in a circular orbit. The interaction energy U is due to the coupling of the induced electric dipole p=(v/c)×μ with the electric field En of the nucleus. According to quantum mechanics, the radius of the electron’s orbit remains constant during a spin-flip transition. Under such circumstances, our model predicts that the energy of the system changes by ΔE=1/2U, the factor 1/2 emerging naturally as a consequence of equilibrium and the change of the kinetic energy of the electron. The correct 1/2 factor for the spin-orbit coupling energy is thus derived without the need to introduce the well-known Thomas precession in the rest-frame of the electron.