Dirac equation in an external field

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

The operator H is intended to describe the energy of a particle in a given external field. As in nonrelativistic quantum mechanics, the influence of an external field is described by a potential-energy V (x) that is added to the kinetic energy Hq. Hence the Dirac operator with an external field reads 

Apart from technical difficulties, there is no principal problem if we let the potential V also depend on time. 

Here q is the electric charge of the particle described by the Dirac equation. Particles in a magnetic field are described by the potential matrix 

This corresponds to the principle of minimal coupling, according to which the interaction with a magnetic field is described by replacing in the Hamiltonian operator the canonical momentum p by the kinetic momentum 11 = p — f A(x). Other types of external-field interactions include scalar or pseudoscalar fields and anomalous magnetic moment interactions. The classification of external fields rests on the behavior of the Dirac equation rmder Lorentz transformations. A brief description of these potential matrices will be given below. 

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Before embarking on the problem of the interaction of the negaton-positon field with the quantized electromagnetic field, we shall first consider the case of the negaton-positon field interacting with an external, classical (prescribed) electromagnetic field. We shall also outline in the present chapter those aspects of the theory of the S-matrix that will be required for the treatment of quantum electrodynamics. Section 10.4 presents a treatment of the Dirac equation in an external field. 

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... 

The noncovariant form of the Dirac equation for an electron in an external field is ... 

For a particle in an external field the situation is significantly more complicated, inasmuch as the terms in the Hamiltonian arising from such a field usually will not commute with the momentum-dependent terms of a transformation such as the one in Eq. [76]. As a result, there is no transformation in closed form that would exactly uncouple the Dirac equation. Instead, one must resort to iterative schemes involving a sequence of approximate transformations with successively smaller coupling between and This approach is very much in the spirit of a perturbation theory, and to some extent one can choose the parameter in which such a perturbation expansion is carried out. Several such methods have been discussed and compared by Kutzelnigg. - ... 

We have here considered the molecular system and the field as one unit descriljed by one Dirac equation. In the usual experimental situation a molecule is somehow placed in an external field, and it is this field that may be controlled for measurements of the system response. In this situation it is natural to divide the total field into internal and external contributions... 

First, we consider the NRF and NRL of the one-electron external field Dirac equation. Before we continue any further, let us rewrite the equation (11), for a fermion in an external radiation field... 

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... 

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... 

In the presence of an external field the Dirac equation will not be invariant, because an external field is not invariant under all Poincare transformations (unless it is a constant). But at least we can expect that the Poincare transformed spinor < (x) — M t/j(A (x — a)) is a solution of the Dirac equation with an appropriately transformed potential matrix Here it has to be assumed that... 

The (time-independent) Dirac equation (DE), which describes a single electron or positron in an external electric field V... 

Before leaving the theoretical formalism section, it is important to note that perturbation theory for relativistic effects can also be done at the fo n-con onent level, i.e. before elimination of the small component by a Foldy-Wouthitysen (FW) or Douglas-Kroll transformation. This is best done with direct perturbation theory (DPT) [71]. DPT involves a change of metric in the Dirac equation and an expansion of this modified Dirac eqtiation in powers of c . The four-component Levy-Leblond equation is the appropriate nonrelativistic limit. Kutzelnigg [72] has recently worked out in detail the simultaneous DPT for relativistic effects and magnetic fields (both external and... 

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. 

According to the Dirac equation for an electron in external electromagnetic fields, Eq. (5.121), we should make the following ansatz for the two-electron system including the external potential energy, Vnuc, of resting nuclei. 

In the presence of an external electromagnetic radiation field the Dirac equation for a fermion takes the form... 

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... 

The effect of the nuclear mass was already mentioned in the introduction. In the Furry picture which is employed in the calculations of QED effects on bound electron states a static external field is assumed which corresponds to an infinitely heavy nucleus. In a non-relativistic approximation its finite mass is encountered by the reduced mass correction similar to the two-body problem in classical mechanics. In a relativistic treatment, however, this approach is oversimplyfied. Recently Artemyev et al. [42, 43] almost solved the whole problem by considering the nucleus as a simple Dirac particle with spin 1 /2, mass M and charge Ze. The interaction of the two Dirac particles electron and nucleus leads to a quasipotential equation in the center-of-mass system,... 

The above result is important because it shows that spin appears naturally in both the Schrddinger and the Dirac equation, but in the former equation the spin is hidden in the term and will only appear on the introduction of an external magnetic field. In other words, spin is not a relativistic effect, as can also be shown by symmetry arguments [55,56]. With A 0 we obtain the equation Wvt = E rWnr with... 

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