Dirac s equation

Spin Particles.—The covariant relativistic wave equation which describes a free spin particle of mass m is Dirac s equation ... 

Gordon s complete solution1) of Dirac s equations. It is seen from figures 5 and 6 that the spin-relativity effect brings the electrons closer to the nucleus. The effect is particularly pronounced for 2 s electrons,... 

No theoretical proof of the Pauli principle was given originally. It was injected into electronic structure theory as an empirical working tool. The theoretical foundation of spin was subsequently discovered by Dirac. Spin arises naturally in the solution of Dirac s equation, the relativistic version of Schrodinger s equation. 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

I returned to the University of Toronto in the summer of 1940, having completed a Master s degree at Princeton, to enroll in a Ph.D. program under Leopold Infeld for which I wrote a thesis entitled A Study in Relativistic Quantum Mechanics Based on Sir A.S. Eddington s Relativity Theory of Protons and Electrons. This book summarized his thought about the constants of Nature to which he had been led by his shock that Dirac s equation demonstrated that a theory which was invariant under Lorentz transformation need not be expressed in terms of tensors. 

G. Breit, Dirac s equation and the spin-spin interactions of two electrons, Phys. Rev. 39 (1932) 616. 

Sadlej AJ, Snijders JG, van Lenthe E, Baerends EJ (1995) Relativistic regular two-component Hamiltonians, four component regular relativistic Hamiltonians and the perturbational treatment of Dirac s equation. J. Chem. Phys. 102 1758-1766... 

There is neither a sharp demarcation in the treatment of solids compared to molecular units, nor is there a strict demarcation in the treatment of different bonding types. It is Schrodinger s (or Dirac s) equation that describes the bonding situation in all cases. Nonetheless, it proves meaningful to use ad hoc classifications. 

Attempts to formulate a causal description of electron spin have not been completely successful. Two approaches were to model the motion on either a rigid sphere with the Pauli equation [102] as basis, or a point particle using Dirac s equation, which is pursued here no further. The methodology is nevertheless of interest and consistent with the spherical rotation model. The basic problem is to formulate a wave function in polar form E = RetS h as a spinor, by expressing each complex component in spinor form... 

Experimental evidence for spin comes from an analysis of atomic line spectra, which show that states with orbital angular momentum (/>0) are split into two levels by a magnetic interaction known as spin-orbit coupling. It occurs in hydrogen but is very small there spin-orbit coupling increases with nuclear charge (Z) approximately as Z4 and so becomes more significant in heavy atoms. Dirac s equation, which incorporates the effects of relativity into quantum theory, provides a theoretical interpretation. 

For materials containing atoms with large atomic number Z, accelerating the electrons to relativistic velocities, one must include relativistic effects by solving Dirac s equation or an approximation to it. In this case the kinetic energy operator takes a different form. 

Whilst this argument mirrors simple textbook derivations for nonrelativistic problems, for example [73, 32], such a non-rigorous treatment is not enough for relativistic calculations. We have already seen in Section 1 that early attempts to solve Dirac s equation by matrix methods encountered unexpected... 

When dealing with relativistic hamiltonians, a great amount of new operators appears because of several approximate procedures allowing the simplification of Dirac s equation. See, for example, the treatise of Bethe and Salpeter [81], as well as the Moss discussion [77 a)], the McWeeny s book [82], or a recent comprehensible brief resume due to Almldf and Gropen [83]. 

In projective relativity the field equations contain, in addition to the gravitational and electromagnetic fields, also the relativistic wave equation of Schrodinger and, as shown by Hoffinann (1931), are consistent with Dirac s equation, although the correct projective form of the spin operator had clearly not been found. The problem of spin orientation presumably relates to the appearance of the extra term, beyond the four electromagnetic and ten gravitational potentials, in the field equations. It correlates with the time asymmetry of the magnetic field and spin. 

Relativistic corrections would take into account the mass increase of the electrons (plus indirect relativistic effects) occurring in heavier atoms (see Section 2.11.1), to be dealt with properly by Dirac s equation, a relativistic theory. But let us concentrate on the above nonrelativistic Hamilton operator and analyze its ingredients. Obviously, H is composed of five parts the first two represent the kinetic energies of the N electrons with masses m and the M nuclei with masses Ma (remember that neither electrons nor nuclei can be expected to be standing still) the differential operator... 

Klein and Nishina (1929) performed a full quantum mechanical calculation using Dirac s equation for the electron. For unpolarized incident photons they obtained for the differential cross section... 

The intensity expressions on p. 12 are derived from non-relativistic quantum mechanics, though relativistic effects are included in the mass m and the electron wavelength A. Starting with Dirac s equation, equation (3) is replaced by... 

Having introduced the principles of special relativity in classical mechanics and electrodynamics as well as the foundations of quantum theory, we now discuss their unification in the relativistic, quantum mechanical description of the motion of a free electron. One might start right away with an appropriate ansatz for the basic equation of motion with arbitrary parameters to be chosen to fulfill boundary conditions posed by special relativity, which would lead us to the Dirac equation in standard notation. However, we proceed stepwise and derive the Klein-Gordon equation first so that the subsequent steps leading to Dirac s equation for a freely moving electron can be better understood. 

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