Transformations exact Foldy-Wouthuysen

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

A. G. Nikitin. On exact Foldy-Wouthuysen transformation. /. Phys. 

The first stages of the procedure are precisely the same as for the exact Foldy-Wouthuysen transformation, which we will follow here. The definitions in the paper by Barysz et al. differ by a sign in the lower row of the transformation matrix, but the results obviously do not depend on this sign. We write the transformation matrix in similar form to the formal unitary Foldy-Wouthuysen transformation ... 

Our line of development here is to return to the exact Foldy-Wouthuysen transformation, presented in section 16.1 and used above to develop a perturbation series. There, we chose ll(2mc -V) as the perturbation and expanded both the square root and the inverse powers. Here, we choose the perturbation parameter as El 2mc - V)... 

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... 

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 ... 

It must be emphasized that these expressions are still exact when compared to the original untransformed operator. The transformed operator /i would be completely decoupled if V = 0, i.e., if the particle were moving freely, and hence only the kinetic energy operator (apart from the rest energy term) remains. Thus, the even terms of the free-particle Foldy-Wouthuysen transformation already account for all so-called kinematic relativistic effects. 

This operator is the two-component analog of the Breit operator derived in section 8.1. The reduction has already been considered by Breit [101] and discussed subsequently by various authors (in this context see Refs. [225,626]). We come back to this discussion when we derive the Breit-Pauli Hamiltonian in section 13.2 in exactly the same way from the free-particle Foldy-Wouthuysen transformation. However, there are two decisive differences from the DKH terms (i) in the Breit-Pauli case the momentum operators in Eq. (12.74) are explicitly resolved by their action on all right-hand side terms, whereas in the DKH case they are taken to operate on basis functions in the bra and ket of matrix elements instead, and (ii) the Breit-Pauli Hamiltonian then results after (ill-defined) expansion in terms of 1 /c (remember the last chapter for a discussion of this issue). 

KB-transformed operators are crucial for the implementation of exact-decoup-ling methods. The free-particle Foldy-Wouthuysen transformation (cf. chapter 11)... 

If n is large (strictly, if it approaches infinity), exact decoupling will be achieved. Usually, a very low value of n is sufficient for calculations of relative energies and valence-shell properties. If the total DKH decoupling transformation, i.e., the product of a sequence of transformations required for the nth order DKH Hamiltonian (without considering the free-particle Foldy-Wouthuysen transformation), is written as... 

The process of making approximations starts with either a partitioning of the Hamiltonian and the metric, as in direct perturbation theory, or, for variational approximations, the elimination of the small component. This can be done either directly or via a Foldy-Wouthuysen transformation. Here, we are interested first and foremost in variational approximations, so we will focus on the elimination of the small component and the Foldy-Wouthuysen transformation. Before considering the approximations, we first outline some theory for the exact solutions. 

The Foldy-Wouthuysen (FW) transformation [67] offers a decoupling, which in principle is exact, but it is impractical and leads to a singular expansion in 1/c in the important case of a Coulomb potential [68]. Douglas and Kroll (DK) suggested an alternative decoupling procedure based on a series of appro-... 

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