Dirac field-free equation

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... 

Solutions of the Dirac equation in field-free space... 

We introduced the field-free nuclear Hamiltonian in section 3.10. Again by analogy with the electronic Hamiltonian, we include the effects of external magnetic fields by replacing P, by P, — Z,eA l in equation (III.248) and the effects of an external electric field by addition of the term Y,a Zae(pa, this treatment is only really justified if the nuclei behave as Dirac particles. The nuclear Zeeman Hamiltonian is thus ... 

The fundamental expressions which describe the interaction of an external magnetic field with the electrons and nuclei within a molecule were developed from the Dirac and Breit equations in chapters 3 and 4. In this section we develop the theory again, making use of the approach described by Flygare [107]. We start with the classical description of the interaction of a free particle of mass m and charge q with an electromagnetic... 

Equations (2.25) and (2.21) do not tell us at which velocity the electron rotates to acquire kinetic and magnetic spin momenta. This is provided by another computation by Dirac [5]. He used a Heisenberg picture with a field-free Hamiltonian (but the conclusion would also hold with a field present) ... 

Lorentz Covariance of the Field-Free Dirac Equation... 

After having derived a truly relativistic quantum mechanical equation for a freely moving electron (i.e., in the absence of external electromagnetic fields), we now derive its solutions. It is noteworthy from a conceptual point of view that the solution of the field-free Dirac equation can in principle be pursued in two ways (i) one could directly obtain the solution from the (full) Dirac equation (5.23) for the electron moving with constant velocity v or (ii) one could aim for the solution for an electron at rest — which is particularly easy to obtain — and then Lorentz transform the solution according to Eq. (5.56) to an inertial frame of reference which moves with constant velocity —v) with respect to the frame of reference that observes the electron at rest. 

Now, we have a unified substitution pattern at hand, which also comprises the time-like coordinates. Substitution of Eq. (5.116) in the field-free Dirac equation as written in Eq. (5.54) yields the covariant form of the Dirac equation with external electromagnetic fields. 

Note that the derivation of the Dirac equation in chapter 5 holds for any freely moving spin-1 /2 fermion and hence also for the proton we come back to such general two-particle Hamiltonians at the beginning of chapter 8 (for an explicit solution of the corresponding two-fermion eigenvalue equation we refer the reader to the work by Marsch [112-115]). In the field-free Dirac Hamiltonian only the rest mass determines which fermion is considered. Accordingly, the total wave function of the hydrogen atom reads... 

Finally, the full Breit-Pauli Hamiltonian contains the one-electron terms discussed in the beginning of this chapter as well as the magnetic interactions of the electrons that can be rewritten as interacting (coupled) angular momenta. Starting from the full Dirac-Breit equation for two electrons, Eq. (8.19), we obtain the purely two-component external-field-free Breit-Pauli Hamiltonian [72, p. 377],... 

While this provides us with an equation for the relativistic electron, the u and matrices arose from the mathematical treatment, and only indirectly from the physics. It would be nice if we could also give these quantities a physical interpretation. In order to find some classical operator or quantity corresponding to a and p, we compare the Dirac Hamiltonian with the classical relativistic Hamiltonian. For this purpose we use the classical relativistic expression for the energy from (2.60) for a field-free system. 

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 relativistic one-fermion equation in the free field presented hy Dirac in 1928 [2,3] is usually written in the contravariant form, which illustrates the relativistic equivalence of space and time in an elegant way... 

Depending on the choice of f + ky, eigenfunctions of the external field Dirac equation ho ext or the free field Dirac equation h, equation (27) represents the Furry bound state molecular Hamiltonian or the free molecular Hamiltonian. 

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. 

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 in an external field is obtained by replacing the free Dirac operator Hq in (3) with an appropriate Dirac operator H,... 

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 above considerations leads to the somewhat troubling question of whether (128) represents the true non-relativistic limit of the Dirac equation in the presence of external fields. Referring back to (110) we have certainly obtained the non-relativistic limit of the free-particle part Lm, but we have in fact retained the interaction term as well as the Lagrangian of the free field. In order to obtain the proper non-relativistic limit, we must consider what is the non-relativistic limit of classical electrodynamics. This task is not facilitated by the fact that, contrary to purely mechanical systems, the laws of electrodynamics appear in different unit systems in which the speed of light appears differently. In the Gaussian system Maxwell s laws are given as... 

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). 

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). 

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 appropriate starting point for the discussion of relativistic Coulomb systems is QED [1,2,3]. The reason for using the full quantum field theory rather than just the Dirac equation is twofold (i) With a quantum field theory the anti-particle sector is sorted correctly. This statement is illustrated by the small table comparing the free particle versions of the two options ... 

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