Current density Dirac

There are correspronding results for Qy,Qz and again the L and S parts differ by two orders of magnitude, the large parts reproducing the densities obtained in 2-component Pauli approximation (cf. (9) et seq.). When the curl of the corresponding magnetization density is added to (43), for the case = 1, we obtain the Dirac current density which already includes the spin term. 

All solutions of this Hamiltonian are thereby electronic, whether they are of positive or negative energy and contrary to what is often stated in the literature. Positronic solutions are obtained by charge conjugation. From the expectation value of the Dirac Hamiltonian (23) and from consideration of the interaction Lagrangian (16) relativistic charge and current density are readily identified as... 

The quantum mechanical expression for the charge-weighted current density is obtained from Eq. (26) when we replace the classical velocity r (f) by the Dirac velocity operator caL and evaluate its expectation value (21),... 

According to the Dirac [36] electron theory, the relativistic wavefunction has four components in spin-space. With the Hermitian adjoint wave function , the quantum mechanical forms of the charge and current densities become [31,40]... 

This result, as well as the form of expressions (23) and (24), shows that the charge and current density relations (3), (4), and (8) of the present extended theory become consistent with and related to the Dirac theory. It also implies that this extended theory can be developed in harmony with the basis of quantum electrodynamics. 

The introduced current density j = So(divE)C is thus consistent with the corresponding formulation in the Dirac theory of the electron, but this introduction also applies to electromagnetic field phenomena in a wider sense. 

Electrons escape from the material by tunnelling through a potential barrier at the surface which has been reduced in thickness to about 1.5 nm by the applied field, Figure 2. If the solid is assumed to contain free electrons which obey Fermi-Dirac statistics, the current density J of field emitted electrons is simply related to the applied field Fand work function cf) by the Fowler-Nordheim (FN) equation... 

We can employ the results of such simulations for both the Dirac and Schitidinger equations in order to calculate the HHG as well as the ATI spectra for the same laser parameters. This allows us to estimate the relativistic effects. An important observable is the multiharmonic emission spectrum S((o). It can be represented as the temporal Fourier transform of the expectation value of the Dirac (SchrOdinger) current density operator j(t) according to... 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

Important for molecular science is how the many-electron probability density and current density can be defined. Following the lines of reasoning introduced in chapter 4, we derive the density and current-density expressions for the many-electron system according to Ehrenfest s theorem employing the many-electron Dirac-Coulomb-(Breit) Hamiltonian,... 

Here, the current density j r) is still required, but it is now uniquely determined by the electron density, j = j[p]. If magnetic interactions between the electrons are not included — that is, when the Dirac-Coulomb Hamiltonian is employed and the Breit or Gaunt terms are omitted — then also the DFT analog of the Coulomb interaction will reduce to a functional J[p] of the density only [396]. Moreover, it now becomes possible to set up theories in which not the full 4-current is used as the fundamental variable, but only those parts of it that are related to the total electron density and the spin density. 

In order to understand the relation of the 4-current to the spin density, it is important to realize that the definition of the current density (naturally) involves a velocity operator, which is in close analogy to classical mechanics (correspondence principle), as we have seen before. As the velocity operator in Dirac s theory contains Dirac matrices a which are composed of Pauli spin matrices cr, we understand that the current density j carries the spin information. 

What is of further concern is whether the probability density is time-independent, which we expect for a bound state, and whether it is conserved under a Lorentz transformation, since this has implications for the normalization of the wave function. If ca is the velocity operator, we may write the current density for the Dirac wave function as... 

Equation (1.6) accounts for both the Arrhenius regime and the temperature-independent low-temperature behavior, as described by the fluctuation-induced tunneling conductivity model. Each of the terms in curly brackets include a description of the forward current density component, in the direction of the applied electric field and a backflow current density in the opposite direction. The first term corresponds to the net current in the low-temperature limit, with an abrupt change in the density of states at the Fermi energy, while the other terms are corrections caused by expansion of the Fermi-Dirac distribution to first order in temperature. 

From the symmetric set, an extended set of Maxwell equations was exhibited in Section V.E. This set contains currents and sources for both fields E, B. The old conjecture of Dirac s is vindicated, but the origin of charge density is always electric (i.e., no magnetic monopole). Standard Maxwell s equations are a limiting case in far field. 

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