Transformations Foldy-Wouthuysen

As described above, the nonrelativistic Pauli wave function consists of two component spinors. It would be of interest to see if the Dirac equation can be reduced to two-component equations since we are specifically interested in the solution to the electronic problem, particularly given that the small component of the Dirac functions are on the order of Z/2c. Furthermore, computational effort is much smaller in two-component solutions than in four-component solutions. The following derivation is due to Greiner.  

Consider first a free particle case. The Dirac equation of a free particle can be written as 

For a free electron, if the S operator is chosen to have the following form in momentum space,  

By expanding e in a Taylor series and collecting the even and odd powers of the expansion terms, one can relate these terms to a trigonometric expansion, such that 

In the presence of a Coulomb potential, there is no exact solution as we found for the free-particle case. By choosing a certain unitary operator, one can systematically reduce the coupling between the large and small components. The Hamiltonian is first decomposed into parts, 

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Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

In the previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

Foldy-Wouthuysen transformation the spin-same orbit h and spin-other orbit... 

Semiclassical methods from quantum mechanics with first-order relativistic corrections obtained from the Foldy-Wouthuysen transformation match with the weak relativistic limit of functionals obtained from quantum electrodynamics, neglecting the (spurious) Darwin terms. 

An explicit expression for the Breit potential was derived in [2] from the one-photon exchange amplitude with the help of the Foldy-Wouthuysen transformation ... 

Hence, at low momenta the photon-nucleus interaction vertex (after the Foldy-Wouthuysen transformation and transition to the two-component nuclear spinors) is described by the expression... 

The Breit-Pauli Hamiltonian is an approximation up to 1/c2 to the Dirac-Coulomb-Breit Hamiltonian obtained from a free-particle Foldy-Wouthuysen transformation. Because of the convergence issues mentioned in the preceding section, the Breit-Pauli Hamiltonian may only be employed in perturbation theory and not in a variational procedure. The derivation of the Breit-Pauli Hamiltonian is tedious (21). 

We have therefore achieved our objective in that equation (3.83), which is correct to order 1 /c2, contains even operators only. It would, of course, be possible to proceed further with the Foldy Wouthuysen transformation but there is little point in doing so, since the theory is inaccurate in other respects. For example, we have treated the electromagnetic field classically, instead of using quantum field theory. Furthermore, we shall ultimately be interested in many-electron diatomic molecules, for which it will be necessary to make a number of assumptions and approximations. 

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... 

In field-free, four-dimensional space, the sequence of Foldy-Wouthuysen transformations can be summed to infinity and written in closed form,... 

In section 3.4 we carried out a Foldy-Wouthuysen transformation on the Dirac Hamiltonian and obtained the result (3.84), correct to order c 2,... 

The first stage in deriving a molecular Hamiltonian is to reduce the Breit equation to non-relativistic form and Chraplyvy [17] has shown how this reduction can be performed by using an extension of the Foldy-Wouthuysen transformation. First let us remind ourselves of the most important features in the transformation of the Dirac Hamiltonian. The latter was written (see (3.57) and (3.58)) as... 

Foldy-Wouthuysen transformation the form of an infinite sum of terms in increasing orders of... 

The weakly relativistic limit of the Hamiltonian (2.20) for fermions in external electric and magnetic fields can be derived with standard techniques, either by direct expansion or by a low order Foldy-Wouthuysen transformation. One obtains... 

Foldy-Wouthuysen transformation, 215 Forbidden reaction, Woodward-Hoffmann... 

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

This defines the exact Foldy-Wouthuysen transformation for the free particle. Note that the square root is not expanded here. 

The spin-orbit force of the Dirac Hamiltonian is obtained by using the Foldy-Wouthuysen transformation as Hg = a r)S.L, where... 

This is not in the Foldy canonical form (23), but can be transformed into it using the Foldy-Wouthuysen transformation [59]... 

The unitary transformation is known in exact analytic form only in the free particle case, when the operation of charge conjugation gives back the same Dirac operator, and is then known as the free-particle Foldy-Wouthuysen transformation [111]. For a general potential, block diagonalization can only be... 

In the nonrelativistic context current-density functional theory is based on the nonrelativistic limits of the paramagnetic current (87) and/or the magnetization density (89) [128,129]. In the relativistic situation, however, a density functional approach relying on jp or m can only be considered an approximation, as long as the external magnetic field does not vanish. In order to clarify the relation between these two points of view the weakly relativistic limit of RDFT has to be analyzed. The weakly relativistic limit of the Hamiltonian (23) can be derived either by a direct expansion in 1/c or by a low order Foldy-Wouthuysen transformation,... 

That the perturbation theory (PT) of relativistic effects has not yet gained the popularity that it deserves, is mainly due to the fact that early formulations of the perturbation expansion in powers of were based on the Foldy-Wouthuysen transformation [11]. In this framework PT is not only formally rather tedious, it also suffers from severe singularities [12, 13], the controlled cancellation of which is only possible at low orders... 

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