Dirac Equation Operator

The Schrodinger equation and the Klein-Gordon equation both involve second order partial derivatives, and to recover such an equation from the Dirac equation we can operate on equation 18.12 with the operator... 

The isomorphism between the tilde operation and hermitian conjugation, implies that upon performing this -operation on the Dirac equation we find that [Pg.524]

The Dirac Equation in a Central Field.—The previous sections have indicated that at times it is useful to have an explicit representation of the matrix element <0 (a ) n> where tfi(x) is the Heisenberg operator satisfying Eq. (10-1). Of particular interest is the case when the external field A (x) is time-independent, Ae = Ae(x), so that the states > can be assumed to be eigenstates of the then... 

Antilinear operator, antiunitary, 688 Antiunitary operators, 727 A-operation, 524 upon Dirac equation, 524 Approximation, 87 methods, successive minimax (Chebyshev), 96 problem of, 52 Arc, 258... 

The electron-electron exchange term, Hex In equation (16) it is necessary to consider only He . As has been discussed, the energy difference between T and S states is equal to Je . With a minimal overlap integral due to a relatively large inter-radical separation. Hex can be given by the Dirac exchange operator [equation (18)],... 

External fields are introduced in the relativistic free-particle operator hy the minimal substitutions (17). One should at this point carefully note that the principle of minimal electromagnetic coupling requires the specification of particle charge. This becomes particularly important for the Dirac equation which describes not only the electron, but also its antiparticle, the positron. We are interested in electrons and therefore choose q = — 1 in atomic units which gives the Hamiltonian... 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

A detailed study of the Dirac equation and its solutions will not be required it will simply be assumed, as already indicated, that the S3rstem of N electrons above the negative-energy sea may be described using a wavefunction constructed from antisymraetrized products of (positive energy) spin-orbitals of type (29). It is, however, necessary to know the basic properties of the operators Q/i, which appear in the Dirac equation... 

For the non-relativistic case (Schrbdinger equation), T = -V. For relativistic case (Dirac equation), T = c a p + 3mc where m is the rest mass of the electron, c is the velocity of light. We have preferred to write the T operator in a general form, covering both cases, given the importance of the relativistic approach in band calculations for actinide solids - see Chap. F... 

In standard quantum field theory, particles are identified as (positive frequency) solutions ijj of the Dirac equation (p — m) fj = 0, with p = y p, m is the rest mass and p the four-momentum operator, and antiparticles (the CP conjugates, where P is parity or spatial inversion) as positive energy (and frequency) solutions of the adjoint equation (p + m) fi = 0. This requires Cq to be linear e u must be transformed into itself. Indeed, the Dirac equation and its adjoint are unitarily equivalent, being linked by a unitary transformation (a sign reversal) of the y matrices. Hence Cq is unitary. 

Abstract. A Dirac equation with hyperfine operator and a recoil structure that remains valid even for positronium is presented and applied to the muonium hyperfine structure to the ordern a8. 

Symmetrically opposite recipes are valid for a Hamiltonian operator in momentum space.) When magnetic fields are present, then the momentum vector receives an additional term, the vector potential A. At relativistic speeds the Dirac equation shall be used. 

For the computational investigation of molecular systems containing heavy atoms, such as transition metals, lanthanides, and actinides, we could neglect neither relativity nor electron correlation. Relativistic effects, both spin-free and spin-orbit, increase with the nuclear charge of atoms. Therefore, instead of the nonrelativistic Schrodinger equation, we must start with the Dirac equation, which has four-component solutions. For many-electron systems, the four-component Hamiltonian is constructed from the one-electron Dirac operator with an approximated relativistic two-electron operator, such as the Coulomb, Breit, or Gaunt operator, within the nopair approximation. The four-component method is relativistically rigorous, which includes both spin-free and spin-orbit effects in a balanced way. However it requires much computational time since it contains more variational parameters than the approximated, one or two-component method. 

Although relativistic effects can be included in the Schrodinger equation by addition of operators describing corrections to the non-relativistic wave function, it is perhaps more satisfying to include relativistic effects by solving the Dirac equation directly. The simplest approximate wave function is a single determinant constracted from four-... 

The Dirac equation is of the same order in all variables (space and time), since the momentum operator p (= — iV) involves a first-order differentiation with respect to the space variables. It should be noted that the free electron rest energy in eq. (8.3) is mc, equal to 0.511 MeV, while this situation is defined as zero in the non-relativistic case. The zero point of the energy scale is therefore shifted by 0.511 MeV, a large amount compared with the binding energy of 13.6eV for a hydrogen atom. The two energy... 

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