Dirac equation electromagnetic theory

Before embarking on the problem of the interaction of the negaton-positon field with the quantized electromagnetic field, we shall first consider the case of the negaton-positon field interacting with an external, classical (prescribed) electromagnetic field. We shall also outline in the present chapter those aspects of the theory of the S-matrix that will be required for the treatment of quantum electrodynamics. Section 10.4 presents a treatment of the Dirac equation in an external field. 

Note that n — N/2 corresponds to the independent particle model analogous to the celebrated Hartree-Fock equations in atomic and molecular physics. We also observe that the fundamental interaction mentioned above is unitarily connected with the electromagnetic interactions between the particle m0 and the antiparticle —m0. Since we do not make any distinctions between the Klein-Gordon and the Dirac equation, we are not able here to integrate the electro-weak theory although in principle this should be possible. 

The behavior of an electron in an electromagnetic held, in the context of the quantum theory, is determined from the solutions of the Dirac equation. Here the free-particle momentum operator is replaced with the generalized 4-momentum operator, pv + e(Av + Bv). The Dirac equation then takes the form... 

It will be shown that the Dirac equation for the tree electron m an external electromagnetic field is leading to the spin concept. Thus, in relativistic theory, the spin angular momentum spears in a natural way, whereas in the non-relativistic formalism, it was the subject of a postulate of quantum mechanics (p. 26). 

The Dirac equation for a particle in the electromagnetic field contains the interaction of the spin magnetic moment with the magnetic field. In this way, spin angular momentum appears in the Dirac theory in a natural way (as opposed to the non-relativistic case, where it has had to be postulated . 

In this section, we derive the two-component Pauli equation from the Dirac equation in external electromagnetic fields. It is also desirable to recover the Schrodinger equation in order to see the connection between relativistic theory and nonrelativistic quantum mechanics. For this purpose, we rewrite Eq. 

In the previous chapter we derived the Dirac equation for an electron moving in an arbitrary electromagnetic potential. We next proceed step-wise toward a many-electron theory which incorporates any kind of electromagnetic interactions. The most simple example is the hydrogen atom consisting of two interacting particles the electron and the proton. 

This book is concerned with the quantum chemical methods for the calculations of electromagnetic properties of molecules. However, in detail only so-called ab initio quantum chemical methods will be discussed in Part III. As ab initio methods one normally describes those quantmn chemical methods that start from the beginning, i.e. methods that require the evaluation of all the terms in the Schrodinger or Dirac equation and do not include other experimentally determined quantities than the nuclear charges, nuclear masses, nuclear dipole and quadrupole moments and maybe positions of the nuclei. This is in contrast to the so-called semi-empirical methods where many of the integrals over the operators in the Hamiltonian are replaced by experimentally or otherwise determined constants. However, in the case of density functional theory (DFT) methods the classification is somewhat debatable. 

Awkward questions about the electromagnetic and gravitational fields of infinitely many particles in the vacuum remain unanswered. Also, the Dirac theory, amended by the hole proposition is certainly not a one-particle theory, and hence not a relativistic generalization of Schrodinger s equation. 

Earlier we mentioned briefly that the electron spin is perfectly consistent with the non-relativistic four-component Levy-Leblond theory [44,45]. The EC type interaction does not manifest in Dirac or Levy-Leblond theory. We shall show that on reducing the four-component Levy-Leblond equation into a two-component form the EC contribution arises naturally. A non-relativistic electron in an electromagnetic radiation field is described by the Levy-Leblond equation given by... 

If there is no explicit external electromagnetic field, the covariant field equations determine a self-interaction energy that can be interpreted as a dynamical electron mass Sm. Since this turns out to be infinite, renormalization is necessary in order to have a viable physical theory. Field quantization is required for quantitative QED. The classical field equation for the electromagnetic field can be solved explicitly using the Green function or Feynman propagator GPV, whose Fourier transform is —gllv/K2, where k = kp — kq is the 4-momentum transfer. The product of y0 and the field-dependent term in the Dirac Hamiltonian, Eq. (10.3), is... 

The fundamental expressions which describe the interaction of an external magnetic field with the electrons and nuclei within a molecule were developed from the Dirac and Breit equations in chapters 3 and 4. In this section we develop the theory again, making use of the approach described by Flygare [107]. We start with the classical description of the interaction of a free particle of mass m and charge q with an electromagnetic... 

To develop the quantum theory of electromagnetic radiation, it is useful to reformulate the harmonic-oscillator problem in terms of creation and annihilation operators, following a derivation due to Dirac. The Schrodinger equation (5.9) can be written... 

The Schrodinger equation was first applied to electromagnetic radiation in 1927 by Paul Dirac [26]. The notion of a quantized radiation field that emerged from this work reconciled some of the apparent contradictions between earlier wave and particle theories of light, and as we will see in Chap. 5, led to a consistent explanation of the spontaneous fluorescence of excited molecules. 

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