Dirac equation free electron

Dirac showed in 1928 dial a fourth quantum number associated with intrinsic angidar momentum appears in a relativistic treatment of the free electron, it is customary to treat spin heiiristically. In general, the wavefimction of an electron is written as the product of the usual spatial part (which corresponds to a solution of the non-relativistic Sclnodinger equation and involves oidy the Cartesian coordinates of the particle) and a spin part a, where a is either a or p. A connnon shorthand notation is often used, whereby... 

For a free electron Dirac proposed that the (time-dependent) Schrodinger equation should be replaced by... 

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... 

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... 

We will describe, in some detail, one such modification, an effective Dirac equation (EDE) which was derived in a number of papers [7, 8, 9, 10]. This new equation is more convenient in many applications than the original BS equation, and we will derive some general formulae connected with this equation. The physical idea behind this approach is that in the case of a loosely bound system of two particles of different masses, the heavy particle spends almost all its life not far from its own mass shell. In such case some kind of Dirac equation for the light particle in an external Coulomb field should be an excellent starting point for the perturbation theory expansion. Then it is convenient to choose the free two-particle propagator in the form of the product of the heavy particle mass shell projector A and the free electron propagator... 

EDE in the external Coulomb field in Fig. 1.6. The eigenfunctions of this equation may be found exactly in the form of the Dirac-Coulomb wave functions (see, e.g, [10]). For practical purposes it is often sufficient to approximate these exact wave functions by the product of the Schrodinger-Coulomb wave functions with the reduced mass and the free electron spinors which depend on the electron mass and not on the reduced mass. These functions are very convenient for calculation of the high order corrections, and while below we will often skip some steps in the derivation of one or another high order contribution from the EDE, we advise the reader to keep in mind that almost all calculations below are done with these unperturbed wave functions. 

Because often only the field-free Pauli Hamiltonian is presented in literature, we shall briefly sketch the derivation of the Hamiltonian hPauh(i) within an external field. For this, we start with the elimination of the small component in the one-electron Dirac equation by substitution of the small component of Eq. (15) to obtain an expression of the large component only... 

The renormalization problem generated by 0(3) is similar to the interaction of the free electron with the vacuum through the Dirac equation [6,15,17] in = 1, h= 1 units ... 

We will discuss at some length the interaction of a free electron with the vacuum, for this is similar to the renormalization problem presented by 0(3)h electrodynamics. An electron interacts with the vacuum according to the Dirac equation... 

Eq. (2) presents the basis for the covariant renormalization approach. The explicit expressions are known for E Ten(E), X u 6 in momentum space. For obtaining these expressions the standard Feynman approach [11,12] or dimensional regularization [13] can be used. They are free from ultraviolet divergencies but acquire infrared divergencies after the renormalization. However, these infrared divergencies, contained in X 1) and cancel due to the Ward identity X -1) = —A1 1 and the use of the Dirac equation for the atomic electron in the reference state a) ... 

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. 

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... 

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... 

The behavior of an electron in an electromagnetic held, in the context of the quantum theory, is determined from the solutions of the Dirac equation. Here the free-particle momentum operator is replaced with the generalized 4-momentum operator, pv + e(Av + Bv). The Dirac equation then takes the form... 

The mathematical basis of the relativistic quantum mechanical description of many-electron atoms and molecules is much less firm than that of the nonrelativistic counterpart, which is well understood. As we do not know of a covariant quantum mechanical equation of motion for a many-particle system (nuclei plus electrons), we rely on the Dirac equation for the quantum mechanical characterization of a free electron (positron) (Darwin 1928 Dirac 1928,1929 Dolbeault etal. 2000b Thaller 1992)... 

The subsequent linearization and quantization of this Hamiltonian by Dirac lead to his celebrated relativistic equation for the free electron [46,47]. 

In this section we shall discuss the interactions that arise upon the introduction of electromagnetic fields in the relativistic electronic Hamiltonian, and we shall also consider the form of electromagnetic interactions in the non-relativistic limit. To simplify matters, we shall first limit our attention to one-electron systems. Consider the time-independent Dirac equation for a free particle... 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

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