Atoms in an electromagnetic field

As illustrated earlier, setting the generator equal to e-p defines the atomic force and the variational principle leads to the integral atomic force law, or the equation of motion for an atom in a molecule. Finally, it was shown that, when F = — sr-p, the commutator defines the electronic kinetic energy and virial for an atom, and the variational principle yields the relationship between these quantities, the atomic virial theorem. These three relationships—the equation of continuity, the equation of motion, and the virial theorem—form the basis for the understanding of the mechanics of an atom in a molecule. 

This chapter concludes with the demonstration that the atomic statement of the principle of stationary action obtains in the presence of an electromagnetic field. Thus, atoms continue to exist in the presence of applied electric and/or magnetic fields and the response of each atom to such fields and its contribution to the magnetic and electric properties of the total system can be determined. 

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The time-dependent Schrodinger equation for an atom in an electromagnetic field reads... 

In order to obtain the Hamiltonian for the system of an atom and an electromagnetic wave, the classical Hamilton function H for a free electron in an electromagnetic field will be considered first. Here the mechanical momentum p of the electron is replaced by the canonical momentum, which includes the vector potential A of the electromagnetic field, and the scalar potential O of the field is added, giving [Sch55]... 

The result for a free electron in an electromagnetic field can be transferred to the Hamiltonian H of an atom by using the same approach. Because the electromagnetic field depends on time, one starts with the time-dependent Schrodinger equation... 

The result states that it is justified to neglect the term A2 in equ. (8.5b) and to treat the interaction between an atom and an electromagnetic field by first-order perturbation theory. The interaction operator is then given by... 

An obvious objection to Weyl s theory is that an atom carried around a closed path in an electromagnetic field would radiate at a different wavelength when reaching the end of the loop. This is refuted by experiment. It was shown by London how to address this problem quantum-mechanically. 

Several articles and book chapters deal with this technique (Srinivasan, 1979). Magnetic nuclear resonance is based on atom properties hydrogen atom nuclei are made of one proton and one neutron and have an overall spin that can be used in NMR. In a pulse-NMR instrument, the molecules are placed in an electromagnetic field, the hydrogen atoms have their nucleus oriented such that their magnetic... 

A similar relation holds for atoms in optical fields where the field frequency (0 is off resonance with the atomic frequency coq. In an electromagnetic field the Lorentz force on an atom in a dilute medium (refractive index n l, - n-l 1) is [13.15]... 

In a plasma, the constituent atoms, ions, and electrons are made to move faster by an electromagnetic field and not by application of heat externally or through combustion processes. Nevertheless, the result is the same as if the plasma had been heated externally the constituent atoms, ions, and electrons are made to move faster and faster, eventually reaching a distribution of kinetic energies that would be characteristic of the Boltzmann equation applied to a gas that had been... 

The Hamiltonian of helium, in the center of mass frame and under the action of an electromagnetic field polarized along the x axis, with field amplitude F and frequency w, reads, in atomic units,... 

Close to this limit the displacements of the two types of atom have opposite sign and the two types of atom vibrate out of phase, as illustrated in the lower part of Figure 8.10. Thus close to q = 0, the two atoms in the unit cell vibrate around their centre of mass which remains stationary. Each set of atoms vibrates in phase and the two sets with opposite phases. There is no propagation and no overall displacement of the unit cell, but a periodic deformation. These modes have frequencies corresponding to the optical region in the electromagnetic spectrum and since the atomic motions associated with these modes are similar to those formed as response to an electromagnetic field, they are termed optical modes. The optical branch has frequency maximum at q = 0. As q increases slowly decreases and... 

We are now able to understand the response of our solid to an electromagnetic field oscillating at frequency >. For the sake of simplicity, we return to the use of expressions (4.17) and (4.18), related to a solid made of single-electron classical atoms, and to only one resonant frequency coq, related to the band gap. Using these expressions, in Figure 4.1(a) we have displayed the dependencies of si and si on the incident photon energy. 

When the fluorescing atoms or molecules are placed inside such a microcavity, the fluorescence gets coupled to the MDR as an electromagnetic field. This results in alternatively enhancement or inhibition of the fluorescence depending on whether or not the fluorescence emission spectrally coincides with a cavity resonance. The effect of MDR on the radiative rate of chelated Europium ions [2] as well as the shortening of fluorescence lifetime of Rhodamine 6G due to the effect of MDR have been reported in microdroplets [3]. 

It is convenient to treat the interaction of atomic electrons with an electromagnetic field in the framework of perturbation theory. 

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