Diracs Theory of the Electron

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

The introduced current density j = So(divE)C is thus consistent with the corresponding formulation in the Dirac theory of the electron, but this introduction also applies to electromagnetic field phenomena in a wider sense. 

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

The aim of this volume is twofold. First, it is an attempt to simplify and clarify the relativistic theory of the hydrogen-like atoms. For this purpose we have used the mathematical formalism, introduced in the Dirac theory of the electron by David Hestenes, based on the use of the real Clifford algebra Cl(M) associated with the Minkwoski space-time M, that is, the euclidean R4 space of signature (1,3). This algebra may be considered as the extension to this space of the theory of the Hamilton quaternions (which occupies an important place in the resolution of the Dirac equation for the central potential problem). 

The usual presentation of the Dirac theory of the electron uses, on the one side, the cy and j, Pauli and Dirac matrices, and on the other side, the Pauli and Dirac spinors. The mathematical language of this presentation is faraway from one of the experiments described in Chap. 2. 

For justifying the form (8.1)-(8.4) of the matrix elements, we will follow the method of perturbation described in the Sects. 29 and 32 of [50]. But here, this method will be directly applied to the Dirac theory of the electron and with the use of the real formalism. 

The way that we follow here differs partially from the one of Schiff [50] but leads to the same conclusion. It is applied here directly to the Dirac theory of the electron instead of the Schrodinger one. 

In order to limit the size of the book, we have omitted from discussion such advanced topics as transformation theory and general quantum mechanics (aside from brief mention in the last chapter), the Dirac theory of the electron, quantization of the electromagnetic field, etc. We have also omitted several subjects which are ordinarily considered as part of elementary quantum mechanics, but which are of minor importance to the chemist, such as the Zeeman effect and magnetic interactions in general, the dispersion of light and allied phenomena, and most of the theory of aperiodic processes. 

VI, 6.7 THE DIRAC THEORY OF THE ELECTRON angular momentum a further angular momentum of magnitude... 

The proof takes different forms in different representations. Here we assume that quantum states are column vectors (or spinors ) iji, with n elements, and that the scalar product has the form ft ip. If ip were a Schrodinger function, J ftipdr would take the place of this matrix product, and in Dirac s theory of the electron, it would be replaced by J fttpdr, iji being a four-component spinor. But the work goes through as below with only formal changes. Use of the bra-ket notation (Chapter 8) would cover all these cases, but it obscures some of the detail we wish to exhibit here. 

For liquid metals, one has to set up density functionals for the electrons and for the particles making up the positive background (ion cores). Since the electrons are to be treated quantum mechanically, their density functional will not be the same as that used for the ions. The simplest quantum statistical theories of electrons, such as the Thomas-Fermi and Thomas-Fermi-Dirac theories, write the electronic energy as the integral of an energy density e(n), a function of the local density n. Then, the actual density is found by minimizing e(n) + vn, where v is the potential energy. Such... 

LS Spin-Orbit Interaction. When a charge moves in an electric field, the theory of special relativity tells us that part of the electric field appears as a magnetic field to the electron. The magnetic moment of the electron interacts with this magnetic field giving rise to what is known as the spin-orbit interaction. From Dirac s theory of the electron, it can be shown (/) that this interaction takes the form... 

In Eq. (15), 8(rik) is the Dirac delta function which, when integrated with the wave function, gives the value of the wave function at rik = 0. The two terms in Eq. (15) are in reality two limiting forms of the same interaction. The first term is the ordinary dipole-dipole interaction for two dipoles that are not too close to each other. It is the proper form of M S1 to be applied to p, d, and / electrons which are not found near the nucleus. For s electrons, which have a finite probability of being at the nucleus, the first term is clearly inappropriate, since it gives zero contribution at large values of rik and does not hold for small values of rik. From Dirac s relativistic theory of the electron, it is found (4) that the second term in Eq. (15) is the correct form for Si when the electron is close to the nucleus. Thus the contribution toJT S] from s electrons is proportional to the wave function squared at the site of the nucleus and the second term in Eq. (15) is often called the contact term in the hyperfine interaction. 

Just as orbital angular momentum L gives rise to a magnetic dipole moment pL, spin angular momentum S gives rise to a spin magnetic dipole moment fis. Dirac s relativistic theory of the electron showed that... 

The next paper was by Dirac on the Theory of the Positron. In the following discussion Niels Bohr made a long intervention on the correspondence principle in connection with the relation between the classical theory of the electron and the new theory of Dirac. 

This attempt to incorporate the spin of the electron, by using a halfintegral quantum number in a theory which seems to require integral values appears very artificial. It does nevertheless agree with the experimental observations. In 1928, Dirac developed a theory of the electron wavefunction which incorporated the principles of Einstein s theory of relativity. Very remarkably, the spin appears as a natural prediction of that theory, although the mathematical details are much too complicated to discuss here. 

The prediction, and subsequent discovery, of the existence of the positron, e+, constitutes one of the great successes of the theory of relativistic quantum mechanics and of twentieth century physics. When Dirac (1930) developed his theory of the electron, he realized that the negative energy solutions of the relativistically invariant wave equation, in which the total energy E of a particle with rest mass m is related to its linear momentum V by... 

Often the work of a first-rate scientist contains more than intended, more than planned, more than anticipated. This was the case with Dirac s theory of the electron unexpected results fell out naturally from his theory. One result of Dirac s work was... 

Physicists were so impressed with Dirac s theory of the electron that they jumped to the conclusion that the same theory would... 

What is the electron The electron, with both particle and wave properties, has four definite, quantitative properties mass, charge, spin, and magnetic moment. Two of these properties, spin and the magnetic moment, seemed to be well accommodated by Dirac theory. (Why the electron has its particular charge and... 

A proper framework for a relativistic description of the various scenarios indicated above is based on Dirac s theory of the electron and QED as the quantum field theory of leptons and photons. 

It is remarkable that the Dirac theory of the relativistic electron perfectly describes this deviation, and the difference to the reference (the nonrelativistic value) is unusually well defined by the limit of a single parameter (the velocity of light) at infinity. The special difficulty encountered in measuring relativistic effects is that relativistic quantum mechanics is by no means a standard part of a chemist s education, and therefore the theory for interpreting a measurement is often not readily at hand. Still, a great many of the properties of chemical substances and materials, in particular, trends across the periodic system of elements, can be understood in terms of relativistic effects without having to consider the details of the theory. 

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