Dirac equation forms

Next we investigate the physical content of the Dirac equation. To that end we inquire as to the solutions of the Dirac equation corresponding to free particles moving with definite energy and momentum. One easily checks that the Dirac equation admits of plane wave solutions of the form... 

Upon noting that u,(p)e ipt and Dirac equation, Eq. (10-213) can also be written in the form... 

For tiie case of a Coulomb field, special methods exist that reduce the equation to a form where the nonrelativistic solutions can be used.11 Thus, if we denote by tji the wave function of the electron moving in a Coulomb field, then t/i obeys the Dirac equation... 

Co-representation matrices explicit forms, 733 multiplication of, 731 of the nonunitary group, 732 Corliss, L. M., 757 Corson, E. M., 498 Coulomb field Dirac equation in, 637 Coulomb gauge, 643,657,664 Counting functions, 165 Covariance matrix, 160 Covariant amplitude of one-particle system, 511 of one, two, etc. particle systems, 511... 

Moreover, instead of describing the electrons by the Dirac equation, that is fully taking into account relativistic effects due to the high acceleration voltage, a modified form of the Schroedinger equation is used, in which electron energy and wavelength are replaced by the equivalent relativistically corrected expressions [85]. 

The solutions of the four coupled Dirac equations (17) must also be solutions of the KG equation and hence have the form of plane waves... 

The Dirac equation is invariant to Lorentz transformations [8], a necessary requirement of a relativistic equation. In the limit of large quantum numbers the Dirac equation reduces to the Klein-Gordon equation [9,10]. The time-independent form of Dirac s Hamiltonian is given by... 

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

However, it is more appropriate to provide theoretical justifications for such use. In this respect, first, we introduce the third category of decoupling of positive and negative states commonly known as the direct perturbation theory . This approach does not suffer from the singularity problems described previously. However, the four-component form of the Dirac equation remains intact. The new Hamiltonian requires identical computational effort as for the Dirac equation itself, hence it is not an attractive alternative to the Dirac equation. However, it is useful to assess the accuracy of approximate two-component forms derived from the Dirac equation such as Pauli Hamiltonian. Consider the transformation... 

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. 

Here we have used the natural expansion (33), with spin-orbitals written in the form (29). The second term in (41), absent in a Pauli-type approximation, contains the correction arising from the use of a 4roomponent formulation it is of order (2tmoc) and is usually negligible except at singularities in the potential. As expected, for AT = 1, (41) reproduces the density obtained from a standard treatment of the Dirac equation but now there is no restriction on the particle number. 

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... 

Eq. (36) may also be expressed as a system of two first-order equations, i.e. as the radial Dirac equation in the representation of Biedenharn. Let us rewrite the radial Dirac-Pauli equation (18) with V = —Zjr in the form... 

One-particle wave-functions in a central field are obtained as solutions of the relativistic Dirac equation, which can be written in the two-component form ... 

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... 

Purely relativistic corrections are by far the simplest corrections to h3rper-fine splitting. As in the case of the Lamb shift, they essentially correspond to the nonrelativistic expansion of the relativistic square root expression for the energy of the light particle in (1.3), and have the form of a series over Zo j /m . Calculation of these corrections should be carried out in the framework of the spinor Dirac equation, since clearly there would not be any hyperfine splitting for a scalar particle. 

In a previous work [33] we suggest an effective approach to study of conditional symmetry of the nonlinear Dirac equation based on its Lie symmetry. We have observed that all the Poincare-invariant ansatzes for the Dirac field i(x) can be represented in the unified form by introducing several arbitrary elements (functions) ( ), ( ),..., ( ). As a result, we get an ansatz for the field /(x) that reduces the nonlinear Dirac equation to system of ordinary differential equations, provided functions ,( ) satisfy some compatible over-determined system of nonlinear partial differential equations. After integrating it, we have obtained a number of new ansatzes that cannot in principle be obtained within the framework of the classical Lie approach. 

Relativity becomes important for elements heavier than the first row transition elements. Most methods applicable on molecules are derived from the Dirac equation. The Dirac equation itself is difficult to use, since it involves a description of the wave function as a four component spinor. The Dirac equation can be approximately brought to a two-component form using e.g. the Foldy-Wouthuysen (FW) transformational,12]. Unfortunately the FW transformation, as originally proposed, is both quite complicated and also divergent in the expansion in the momentum (for large momenta), and it can thus only be carried out approximately (usually to low orders). The resulting equations are not variationally stable, and they are used only in first order perturbation theory. 

On this empirical evidence, it is possible to reach a far-reaching conclusion that all wave functions in quantum mechanics are of the form (590). For example, the electron wave function from the Dirac equation is... 

These laws are useful but represent cause without effect, that is, fields propagating without sources, and the Maxwell displacement current is an empirical construct, one that happens to be very useful. These two laws can be classified as U(l) invariant because they are derived from a locally invariant U(l) Lagrangian as discussed already. Majorana [114] put these two laws into the form of a Dirac-Weyl equation (Dirac equation without mass)... 

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