The Dirac equation for a free electron

The quadratic form of the relativistic Hamilton function is inappropriate for its operator form because the second derivative with respect to time is involved 

Consequently the integral of l l2 over all space varies with time so that the probability is not conserved. 

The equation linear in time derivatives postulated by Dirac [1] is 

This expression is identical to the Klein-Gordon equation under the assumption that the following relationships (idempotency and anticommutation) hold true 

The operator a0 is identity. Because of the anticommutation relations, the Dirac operators cannot be multiplicative operators (numbers). They are not differential operators either because of the independence of px,Py,pz,Po,x,y,z,t. But what variable (degree of freedom) do the Dirac operators act upon In the chapter dealing with the electron spin we saw that there are the Pauli matrix operators (which obey the idempotency and anticommutation) acting on a two-component wave function (the two-component spinor) 

---

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... 

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

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) ... 

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 purpose of this section is to show how the problem of passing from the four-component Dirac equation to two-component Pauli-like equations can be systematically investigated within the framework of the theory of effective Hamiltonians.Beyond the above-mentioned difficulties, we will be able to derive energy-independent two-component effective Hamiltonians that can be used for variational atomic and molecular calculations. To introduce the subject and the notation, let us first consider the simple case of a free electron. 

It will be shown that the Dirac equation for the free electron in an external electromagnetic field is leading to the spin concept. Thus, in relativistic theory, the spin angular momentum appears in a natural way, whereas in the non-relativistic formalism it was the subject of a postulate of quantum mechanics, p. 25. 

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... 

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... 

One can gain some insight into the nature of the Dirac wave equation and the spin angular momentum of the electron by considering the Foldy Wouthuysen transformation for a free particle. In the absence of electric and magnetic interactions, the Dirac Hamiltonian is... 

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. 

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 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). 

Up of quarks), electrons, and neutrinos. In addition to the electron, the only stable and isolatable particles are the proton and the neutrino. One may add the neutron, which decomposes into a proton, an electron, and an antineutrino when isolated. As the Dirac equation in free space does not refer to the charge (or the stability) of the electron, the only conditions for other particles to obey this equation are to have a rest mass and spin Vi. All that was said for the electron should then hold for these three particles, as well as for the others in the electron and neutrino families. 

Paul Dirac used the Fock-Klein-Gordon equation to derive a Lorentz transformation invariant equation for a single fermion particle. The Dirac equation is solvable only for several very simple cases. One of them is the free particle (Dirac), and the other is an electron in the electrostatic field of a nucleus (Charles Darwin-but not the one you are thinking of). 

---