Dirac continuity equation

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] ... 

Again, the summation convention is used, unless we state otherwise. As will appear below, the same strategy can be used upon tbe Dirac Lagrangean density to obtain the continuity equation and Hamilton-Jacobi equation in the modulus-phase representation. 

Eq. (7.8) is the most general covariant form of the inhomogeneous Maxwell equations, which immediately imply the continuity equation dy.j = + div = 0 of section 5.2.3, and Eq. (7.9) is the covariant time-dependent Dirac equation in the presence of external electric and magnetic fields. The homogeneous Maxwell equations are automatically satisfied by the sole existence of... 

What is the relation between the lORA energies and the ZORA and Dirac energies There is a correspondence at =0 and we expect that the correspondence continues in the vicinity of this point. Unlike the ZORA equation, we cannot perform a scaling to obtain a relation with the Dirac ESC equation, and therefore we cannot obtain a direct relation with the Dirac eigenvalues. What we can do is to make use of the Rayleigh quotient for (18.37) to obtain a relation between the ZORA and lORA eigenvalues, since ZORA and lORA have the same Hamiltonian but a different metric. For an arbitrary wave function r] . 

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). 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

First, we consider the NRF and NRL of the one-electron external field Dirac equation. Before we continue any further, let us rewrite the equation (11), for a fermion in an external radiation field... 

Then Y is an element of a non-standard sequence if Y e R and M e N. We say that YM converges to Y if Y - st Y for all infinite integers M. Continuity and differentiability can be similarly defined, for example ff(a) = st (f(a+h) - f(a) /h for all infinitesimals h. The Dirac delta function can be defined pointwise and new approaches to the theory of probability and stochastic differential equations are opened up. But these are applications and paradoxically I have been describing the abstraction that is the non-standard world in very concrete terms. This is no place to go back and try to present it abstractly, but it stands as an example of a recent step forward in the path of abstraction that began two and a half millenia ago (see also (23,24)). 

The LDA radial Schrodinger equation is solved by matching the outward numerical finite-difference solution to sin inward-going solution (which vanishes at infinity) of the same energy, near the classical turning point. Continuity of P t(r) = rAn/(r) and its derivative determines the eigenvalue /. The second order differential equation is actually solved as a pair of simultaneous first-order equations, so that the nonrelativistic and relativistic (Dirac equation) procedures appear similar. 

During the last few decades, general computer programs, based on the Dirac equation, for extensive four-component electronic structure calculations for atoms have been developed by several groups. These programs are being improved and extended continually, incorporating the explicit treatment of the Breit interaction as well as an ever more sophisticated consideration of interelectron correlation and even of QED effects. 

In perturbation theory, one often considers the essential spectrum. It consists of the continuous spectrum together with the accumulation points of the eigenvalues. Whenever the potential vanishes at infinity, the essential spectrum of the Dirac equation is the set... 

The limiting cases of continuous reactors considered in most reactor design textbooks are the perfectly mixed stirred tank and the plug-flow tube. These reactors can differ significantly in the amount of mixing and, therefore, the residence time distribution. The plug-flow tube (PFT) is assumed to be without any axial mixing. Hence, at steady state, the residence time distribution of the material in the effluent stream is represented by the Dirac function as shown by Equation (8.1) ... 

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