Relativistic quantum mechanics

In the non-relativistic quantum mechanics discussed in this chapter, spin does not appear naturally. Although... 

A marvellous and rigorous treatment of non-relativistic quantum mechanics. Although best suited for readers with a fair degree of mathematical sophistication and a desire to understand the subject in great depth, the book contains all of the important ideas of the subject and many of the subtle details that are often missing from less advanced treatments. Unusual for a book of its type, highly detailed solutions are given for many illustrative example problems. 

The central equation of (non-relativistic) quantum mechanics, governing an isolated atom or molecule, is the time-dependent Schrodinger equation (TDSE) ... 

The aim of this section is to show how the modulus-phase formulation, which is the keytone of our chapter, leads very directly to the equation of continuity and to the Hamilton-Jacobi equation. These equations have formed the basic building blocks in Bohm s formulation of non-relativistic quantum mechanics [318]. We begin with the nonrelativistic case, for which the simplicity of the derivation has... 

W. Greiner, Relativistic Quantum Mechanics Wave Equations, Springer-Verlag, Berlin, 1997. 

Correlated and relativistic quantum mechanical calculations give the highest possible accuracy and are necessary for heavy atoms or correlation-sensitive systems. 

Also arising from relativistic quantum mechanics is the fact that there should be both negative and positive energy states. One of these corresponds to electron energies and the other corresponds to the electron antiparticle, the positron. 

The most common description of relativistic quantum mechanics for Fermion systems, such as molecules, is the Dirac equation. The Dirac equation is a one-electron equation. In formulating this equation, the terms that arise are intrinsic electron spin, mass defect, spin couplings, and the Darwin term. The Darwin term can be viewed as the effect of an electron making a high-frequency oscillation around its mean position. 

Dirac equation one-electron relativistic quantum mechanics formulation direct integral evaluation algorithm that recomputes integrals when needed distance geometry an optimization algorithm in which some distances are held fixed... 

Bas, A.I., Ya.B. Zeldovich and A.M. Perelomov, 1971, Scattering Reactions and Decays in the Non-relativistic Quantum Mechanics (Nauka, Moscow [in Russian)]. 

There is a nice point as to what we mean by the experimental energy. All the calculations so far have been based on non-relativistic quantum mechanics. A measure of the importance of relativistic effects for a given atom is afforded by its spin-orbit coupling parameter. This parameter can be easily determined from spectroscopic studies, and it is certainly not zero for first-row atoms. We should strictly compare the HF limit to an experimental energy that refers to a non-relativistic molecule. This is a moot point we can neither calculate molecular energies at the HF limit, nor can we easily make measurements that allow for these relativistic effects. 

Historical Background.—Relativistic quantum mechanics had its beginning in 1900 with Planck s formulation of the law of black body radiation. Perhaps its inception should be attributed more accurately to Einstein (1905) who ascribed to electromagnetic radiation a corpuscular character the photons. He endowed the photons with an energy and momentum hv and hv/c, respectively, if the frequency of the radiation is v. These assignments of energy and momentum for these zero rest mass particles were consistent with the postulates of relativity. It is to be noted that zero rest mass particles can only be understood within the framework of relativistic dynamics. 

We have thus far only considered the relativistic quantum mechanical description of a single spin 0, mass m particle. We next turn to the problem of describing a system of n such noninteracting spin 0, mass m, particles. The most concise description of a system of such identical particles is in terms of an operator formalism known as second quantization. It is described in Chapter 8, The Mathematical Formalism of Quantum Statistics, and Hie reader is referred to that chapter for detailed exposition of the formalism. We here shall assume familiarity with it. 

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