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CRC Press eBooks

Interaction of Electrons with Electromagnetic Field

Şaki̇r Erkoç

Published Jul 28, 2006• 0 citations
Publisher

Abstract

Hydrogen atom may be modeled as the interaction of an electron with the static Coulomb field due to a point charge. One may generalize this to the interaction with an external magnetic or electric field. Before going into the details it is better to review the classical theory of electromagnetism.
DOI
10.1201/9781584887331-10

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The hydrogen atom with the Coulomb interaction is one of the exactly solvable non-relativistic quantum models. Unlike many other exactly solvable models it describes a real physical object providing the formulas for energy levels and stationary state wave functions of a real hydrogen atom. In this paper we modify the model replacing the Coulomb interaction by the interaction of the proton and the electron with the classical electromagnetic field serving as an intermediary transmitting the electromagnetic interaction of these two charged quantum particles.
Abstract The interaction between atoms or molecules and electromagnetic (EM) fields underlies all spectroscopic techniques and a great variety of desirable molecular properties. EM fields and EM waves are introduced via the famous Maxwell equations. The scalar and vector potential are defined, and the gauge freedom is outlined. The Coulomb gauge is adopted. The classical Hamiltonian for a charged particle in an EM field is derived, and from this the ‘minimal substitution’ rules for incorporating the fields in the quantum Hamiltonian are obtained. The operators describing the interaction of an electron with static electric and magnetic fields, including the magnetic fields from nuclear spins, are derived, followed by the derivation of the interaction between an electron and an EM wave.
We begin our study of atomic physics by relying on the phenomenological notions summarized in chapter 1 and by fully exploiting the quantum mechanical formalism outlined in chapter 2. A hierarchy of approximations will be introduced in the first instance to describe the hydrogen atom, which allows one to focus on the main features of its energy spectrum and quantum properties, including the key ‘atomic orbital’ concept. Next, these results will be extended to the more general case of hydrogenic atoms. The physical picture will be made more rigorous by introducing the magnetic interactions, included those linked to the intrinsic spin degree of freedom, leading to the full description of the fine and hyperfine structure of one-electron atoms. The effects of a static external magnetic or electric field will be eventually discussed by calculating the corresponding energy shifts and splittings of the energy levels.
1. Introduction. Atomic Processes in Astrophysics 2. Radiative Transitions in an Atomic Potential 3. Polarization of Atoms. Polarization Radiation and Light Scattering 4. Atom in a Static Electric Field 5. Atom in a Magnetic Field and Crossed F-B Fields 6. Atom in a Variable Electric Field 7. Atom under Influence of Stochastic Peturbations 8. Relaxation and Decay of Atomic States 9. Excited Hydrogen-Like Atom in Electric Fields of Charged Particles. Foundation of Classical and Quantum Spectral Line Broadening Theory 10. Quasiclassical Theory of Atom - Charge Particle Collisions 11. Collision of the Atom with Atomic Particles in External Fields 12. Atomic Kinetics References Index
The world should have waited until 1913, when a Danish physicist N. Bohr proposed an entirely new theory for a description and partial explanation of the spectral properties of the hydrogen atom. His theory was based on two postulates, the so-called Bohr postulates, and both of them were impossible to be understood from the point of view of classical physics. Being based on these postulates and employing classical physics, N. Bohr was able to derive explicit expressions for the energy levels of the hydrogen atom and the wavelengths of possible transitions. The hydrogen-like atom has only one electron, and all others have been removed by various ways of ionization. An essential feature of these systems is the presence of one electron only. The Coulomb interaction between this electron and the nucleus is spherically symmetrical. Later, we shall see that the situation becomes considerably more complicated in the case of multielectron atoms.
It is shown that both the electric and magnetic dipole moment vectors of hydrogen atom in the excited states with wave function un(±)=12[Rn,n−1(r)Yn−1,±(n−2)(θϕ)±Rn,n−2(r)Yn−2,±(n−2)(θϕ)] u_n^{(\pm)} = {1\over\sqrt 2} [R_{n,n-1}(r) Y_{n-1,\pm (n-2)}(θϕ) \pm R_{n,n-2}(r) Y_{n-2,\pm (n-2)}(θϕ)]un(±)​=2​1​[Rn,n−1​(r)Yn−1,±(n−2)​(θϕ)±Rn,n−2​(r)Yn−2,±(n−2)​(θϕ)] align themselves in the direction of an external uniform electric field which is characteristic of magneto-electric effect. These states are found to have magnetic charge g=3n(n−2)eg={3n\over (n-2)e}g=(n−2)e3n​ on account of this effect. This result is confirmed by an independent method. An experiment is suggested to fabricate these states and detect the magnetic charge. It ma be worth noting that inspite of many experimental searchs, magnetic charge, whose existence has been theorized both in electrodynamics and non-abelian gauge theories, none have been found so far, nor there exist any suggstion as to where these are to be found.