The hydrogen-like atom in Dirac theory

Despite the glorious invariance with respea to the Lorentz transformation and despite spectacular successes, the Dirac equation has some serious drawbacks, including a lack of clear physical interpretation. These drawbacks are removed by a more advanced theory - quantum electrodynamics. 

instead of the non-relativistic energy equal to from the Dirac theory we obtain —0.5000067 a.u., which means a very small correction to the non-relativistic energy. The electron energy levels for the non-relativistic and relativistic cases are shown schematically in Fig. 3.2. 

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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 quantum number k will turn out to play an analogous role in the Dirac theory of atoms that is played by I in the nonrelativistic theory (i.e., being associated with a constant of the motion). In the 1930s, Swirles introduced a special notation to assign the quantum numbers of different quantum mechanical states Yj(r) to the familiar spectroscopic notation for the states of hydrogen-like atoms such as s=sharp, p=principal, d=diffuse (see Table 6.1 for an overview). 

Many attempts have been undertaken to rewrite the one-electron Dirac equation — of hydrogen-like atoms and also the mean-field SCF type derived in chapter 8 and in matrix form in chapter 10 — to obtain a form that is most suitable for numerical computations. Historically, the transformation and elimination techniques first emerged from such endeavors and were only later studied from a formal point of view as an essential part of the complete picture of relativistic many-electron theory. For instance, the DKH theory was first developed as an efficient low-order approximation to the Dirac equa-... 

Like Bohr s model of the hydrogen atom, Sommerfeld s theory flowered only briefly. The creation of quantum mechanics and the discovery of electron spin, both in 1925, followed by Paul Dirac s theory in 1928, provided a solid theory-based underpinning for... 

The hydrogen atom is the most important testing ground for any quantum mechanical theory. Amazingly, it turns out that the Dirac equation describing a hydrogen-like system can be solved analytically. The results are in almost perfect agreement with the measurements. This success has been one of the main historical reasons for the quick acceptance of the Dirac equation by physicists. 

Similar correction cases, either in higher orders or on excited hydrogenic states, may be considered following the same line of analysis however with the same principally conclusion that they do not in fact contribute to the real shift (or perturbation) of the hydrogenic atoms within the point like nucleus framework this leads with the idea that indeed, for atomic and supra-atomic systems the point like hypothesis of the nucleus finely works and will be in next assumed as such (for instance when treating the chemical bonding by means of the Dirac theory, see next chapters). 

This classic text is the standard reference for the Dirac theory of hydrogen- and helium-like atoms. Bethe had already published a review article on one- and two-electron atoms in Handbuch der Physik in 1933 [72], which is very readable — especially because it demonstrates the early difficulties with a quantum theory of boimd-state electrons and their interactions. 

Indeed, what they disparagingly referred to as the second quantum theory appeared less like an attempt at reducing chemistry to physics than an attempt to redefine the latter in conformity with the former. But Dirac s fellow physicists were not too concerned either, as his project failed to address the real issue of establishing procedures for calculation and approximation. These problems needed to be overcome before anyone could offer analytic quantum mechanical treatments of any atoms or molecules apart from hydrogen and helium, let alone their interactions. 

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