The Foldy-Wouthuysen transformation

We start from the time-independent Dirac equation in two-component form. 

Here we are not considering the time dependence, which was done in the original paper by Foldy and Wouthuysen (1950). It turns out that the results of the transformation are essentially the same. We are looking for a transformation that transforms a 4-spinor to a 2-spinor, such that 

This means that the lower row elements of Q must fulfill the relation 

To achieve this, we introduce the operator X, which connects and This relation indicates that a suitable form of Q might be 

To make this transformation unitary, we determine the norm from 

Although we have already presented, in sections (2.3, 2.5, 2.6), three main ways from the DE to the SE as its nrl, we must now discuss a fourth one, which has played an important historical role, although it can now be regarded as obsolete [12, 13], namely that via the Foldy-Wouthuysen (FW) transformation [11]. At first glance, this looks even more elegant than the three ways reported previously. While in these one had to specify whether one considers electronic or positronic states, before one performed the limit, by means of the FW transformation the two limits are obtained in a single shot. 

We proceed somewhat similarly as in the theory of effective Hamiltonians (see the Appendix). However, we do, of course, not consider the matrix representation of the Dirac operator in a given basis, but take directly the matrix form of D in terms of the upper and lower spinor components. 

A closed expression of this transformation can be found for a free particle, 

A convergent expansion in powers of c is only possible if we restrict the domain of H to non-ultrarelativistic states, for which 

Note that eigenstates of D lare also eigenstates of p, i.e. for a free particle p is a constant of motion. Even for a free particle an expansion of the transformed Hamiltonian (after removal of the rest energy) is only possible in a very limited sense. There is divergence for ultrarelativistic states. The nrl is obviously 

In section 3.2 we pointed out that the Dirac Hamiltonian contains operators which connect states of positive and negative energy. What we now seek is a Hamiltonian which is relativistically correct but which operates on the two-component electron functions of positive energy only. We require that this Hamiltonian contain terms representing electromagnetic fields, and Foldy and Wouthuysen [12] showed, by a series of unitary transformations, that such a Hamiltonian can be derived. The Dirac Hamiltonian 

Before proceeding we must state a few important properties of the operators S and 0. These are 

Hence f is an eigenfunction satisfying the transformed Schrodinger equation 

The transformed Hamiltonian X can be written in a more useful form by expanding the exponentials in (3.66), i.e. 

Now Foldy and Wouthuysen suggested that if Xis the Dirac Hamiltonian (3.58), the operator S should be given by 

The transformed Hamiltonian X can be written in a more useful form by expanding 

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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... 

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... 

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... 

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... 

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]... 

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... 

Of course, the Dirac operator for H-like ions has continuum states as well, including ultrarelativistic ones. One can therefore neither expect that all eigenstates are analytic in c , nor that the entire Dirac operator allows an expansion in powers of c. This can at best be the case for the projection of D to positive-energy non-ultrarelativistic states. The paradigm, on which the Foldy-Wouthuysen transformation is based, to construct a Hamiltonian, related to the Dirac operator by a unitary transformation, in an expansion in c ... 

PERTURBATION THEORY BASED ON THE FOLDY -WOUTHUYSEN TRANSFORMATION... 

Explicit inclusion of relativistic effects in valence-only calculations has been by far less frequently attempted. Datta, Ewig and van Wazer [135] used a Phillips-Kleinman PP in a study of PbO, whereas Ishikawa and Malli [136] tested PPs of semilocal form in four-component atomic DHF finite difference calculations. This work was extended by Dolg [137] to four-component molecular DHF calculations with a subsequent correlation treatment. In addition a complicated form of Vcv based on the Foldy-Wouthuysen transformation [138] was derived by Pyper [139] and applied in atomic calculations [140]. For all these approaches the computational effort is significantly higher than for the implicit treatment of relativity, and the gain of computational accuracy is not obvious at all. 

Although the Dirac equation in its four-component form can be solved exactly for a few systems (the hydrogen-like atom, electron in a uniform magnetic field), normally a decoupling to the two-component form has to be done [2-6], For this purpose two techniques were developed the Foldy-Wouthuysen transformation and the partitioning method (for a small component). We will follow the second approach, which is based on these steps ... 

One seemingly sensible approach to the relativistic electronic structure theory is to employ perturbation theory. This has the apparent advantage of representing supposedly small relativistic effects as corrections to a familiar non-relativistic problem. In Appendix 4 of Methods of molecular quantum mechanics, we find the terms which arise in the reduction of the Dirac-Coulomb-Breit operator to Breit-Pauli form by use of the Foldy-Wouthuysen transformation, broken into electronic, nuclear, and electron-nuclear effects. FVom a purely aesthetic point of view, this approach immediately looks rather unattractive because of the proliferation of terms at the first order of perturbation theory. To make matters worse, many of the terms listed are singular, and it is presumably the variational divergences introduced by these operators which are referred to in [2]. Worse still, higher-order terms in the Foldy-Wouthuysen transformation used in this way yield a mathematically invalid expansion. 

Another major two-component approximation is the Foldy-Wouthuysen transformation (Foldy and Wouthuysen 1950), which makes the large-component and small-component submatrices of the Dirac Hamiltonian matrix, Hd, linear independent by a unitary transformation such as... 

The effect of small components on the properties of molecules have been studied by Schwarz to a high order using the Foldy-Wouthuysen transformation. Schwarz has demonstrated that the contribution of the small components to chemical properties can be ignored. Thus one can ignore the small components if one is considering chemical properties. Examples of four-component atomic spinors are shown in the review paper by Pitzer. ... 

Several authors have considered a number of approximate solutions to the Dirac equation. One such method is the use of the Foldy-Wouthuysen transformation (see, for example, Morrison and Moss ). Upon application of a unitary transformation of the form... 

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