Relativistic wave equation

For a detailed discussion of the merits and limitations of a potential approach and of its aspects (wave equations, relativistic effects, flavour dependence and ambiguities connected with spin dependence), see Licht-enberg et al. (1988 and 1989). 

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

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

Actually Schrddinger s original paper on quantum mechanics already contained a relativistic wave equation, which, however, gave the wrong answer for the spectrum of the hydrogen atom. Due to this fact, and because of problems connected with the physical interpretation of this equation, which is of second order in the spaoe and time variables, it was temporarily discarded. Dirac took seriously the notion of first... 

For the connection between group theory and relativistic wave equations, see ... 

Corson, E. M., Tensors, Spinors and Relativistic Wave Equations, Hafher Publish ing Co., New York, 1953. 

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

Spin 1, Mass Zero Particles. Photons.—For a mass zero, spin 1 particle, the set of relativistic wave equations describing the particle is Maxwell s equations. We adopt the vector 9(x) and the pseudovector (x) which are positive energy (frequency) solutions of... 

Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A. M. Dirac (1928) developed a quantum mechanics based on the theory of relativity rather than on Newtonian mechanics and applied it to the electron. He found that the spin angular momentum and the spin magnetic moment of the electron are obtained automatically from the solution of his relativistic wave equation without any further postulates. Thus, spin angular momentum is an intrinsic property of an electron (and of other elementary particles as well) just as are the charge and rest mass. 

In order to preserve the resemblance to Schrodinger s equation Dirac obtained another relativistic wave equation by starting from the form... 

The relativistic (DSW) version incorporates the same approximations but starts from the Dirac rather than the Schroedinger wave equation,(11)... 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

After the discovery of the relativistic wave equation for the electron by Dirac in 1928, it seems that all the problems in condensed-matter physics become a matter of mathematics. However, the theoretical calculations for surfaces were not practical until the discovery of the density-functional formalism by Hohenberg and Kohn (1964). Although it is already simpler than the Hartree-Fock formalism, the form of the exchange and correlation interactions in it is still too complicated for practical problems. Kohn and Sham (1965) then proposed the local density approximation, which assumes that the exchange and correlation interaction at a point is a universal function of the total electron density at the same point, and uses a semiempirical analytical formula to represent such universal interactions. The resulting equations, the Kohn-Sham equations, are much easier to handle, especially by using modern computers. This method has been the standard approach for first-principles calculations for solid surfaces. 

Qualitatively, internal orbitals are contracted towards the nucleus, and the Radon core shrinks and displays a higher electron density. External electrons are much more efficiently screened from the nucleus, and the repulsion described by the (Hartree-Fock) central potential U(rj) in Eq. (7) of the preceding subsection is greatly enhanced. In fact, wave functions and eigenstates of external electrons calculated with the relativistic and the non-relativistic wave equations differ greatly. 

The so-called Hartree-Fock (HF) limit is important both conceptually and quantitatively in the quantum mechanical theory of many-body interactions. It is based upon the approximation in which one considers each particle as moving in an effective potential obtained by averaging over the positions of all other particles. The best energy calculated from a wavefunction having this physical significance is called the Hartree-Fock energy and the difference between this and the exact solution of the non-relativistic wave equation is called the correlation energy. 

Bhabha treated the general relativistic wave equations and Tonnelat presented the idea of Louis de Broglie of trying to describe a photon as an object composed of two neutrinos.63... 

P. Gueret and J. P. Vigier, Relativistic wave equations with quantum potential nonlinearity, Lett. Nuovo Cimento 38, 125—128 (1983). 

In this section we discuss the nonrelativistic 0(3) b quantum electrodynamics. This discussion covers the basic physics of f/(l) electrodynamics and leads into a discussion of nonrelativistic 0(3)h quantum electrodynamics. This discussion will introduce the quantum picture of the interaction between a fermion and the electromagnetic field with the magnetic field. Here it is demonstrated that the existence of the field implies photon-photon interactions. In nonrelativistic quantum electrodynamics this leads to nonlinear wave equations. Some presentation is given on relativistic quantum electrodynamics and the occurrence of Feynman diagrams that emerge from the B are demonstrated to lead to new subtle corrections. Numerical results with the interaction of a fermion, identical in form to a 2-state atom, with photons in a cavity are discussed. This concludes with a demonstration of the Lamb shift and renormalizability. 

The galactic redshift could obviously be attributed to the damping of the electromagnetic waves emitted from various galaxies in random motion within a stationary universe. Now, comparison between Hubble relativistic linear law and the logarithmic law that comes out from Maxwell electromagnetic wave equation shows that, in any case, the logarithmic law fits experimental data very well and thus better than linear law. 

Several authors have found additional solutions to the relativistic wave equations. These solutions can be summarized as follows ... 

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... 

In non-relativistic perturbative atomic Z-expansion theory, as recently sum-marked [11], a new scaled length, p = Zr, and a scaled energy, e= T2 E, are introduced in the many-electron wave equation. That is, the units of length and energy are changed to 1/Z and Z2 a.u., respectively. The Hamiltonian then takes the form... 

The simplest derivation , given in many books, e.g. in chapter 4, was in fact similar to that used by Schrodinger to obtain an equation which falls short of the relativistic Schrodinger equation only by the absence of spin, a concept which had not yet arisen [1], This first quantum-mechanical wave equation is now known as the Klein-Gordon equation, and applies to particles without spin. 

Schrodinger, Heisenberg. Non-relativistic quantum mechanics. Wave equation yields Bohr energy levels. 

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