Foldy-Wouthuysen Expansion in Powers of

Every transformation method aiming at decoupling of the Dirac Hamiltonian (now written with the energy level shift of —ntgC from section 6.7), 

Historically, 1/c expansions, which allow one to approach the well-known nonrelativisitic limit via c — oo, were the only ones seriously considered for decades. Therefore, we strictly refer to 1/c expansions as Foldy-Wouthuysen expansions. This is in accordance with the original intention of Foldy and Wouthuysen in Ref. [609]. Expansions in terms of the scalar potential V are called Douglas-Kroll-Hess (DKH) expansions. 

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At this stage we should add the missing extemal-field-dependent operators to the Breit-Pauli Hamiltonian reviewed already by Bethe [72]. By contrast to what follows, these terms are also derived in the spirit of the ill-defined Foldy-Wouthuysen expansion in powers of 1 /c. However, since the molecular property calculation is carried out in a perturbation theory anyhow, we may utilize the complete field-dependent Breit-Pauli Hamiltonian in such calculations. 

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 ... 

One issue raised by the use of an expansion in powers of the potential is that of electric gauge invariance. If we add a constant to the potential, we should obtain a constant shift in the energy, if the potential is gauge-invariant. Terms that are of second order in the potential would be expected to give rise to a quadratic term in the added constant. Looking at the free-particle Foldy-Wouthuysen transformed Hamiltonian, (16.42), it is apparent that the added constant only survives in the even operator the odd operator involves a commutator that eliminates the constant. Consequently, the... 

The complicated nature of the operator VVi raises the question of whether a simpler alternative can be found. Barysz, Sadlej, and Snijders (1997) devised an approach which starts from the free-particle Foldy-Wouthuysen transformation, just as the Douglas-Kroll transformation does. Whereas the Douglas-Kroll approach seeks to eliminate the lowest-order odd term from the transformed Hamiltonian, their approach seeks to be correct to a particular order in 1 /c, and it provides a ready means for defining a sequence of approximations of increasing order in 1/c. It is important to note that, while the expansion in powers of 1 /c that resulted from the Foldy-Wouthuysen transformation in section 16.2 generates highly singular operators, this is not true per se of expansions in 1/c. What multiplies 1/c is all-important. In the case of the Foldy-Wouthuysen transformation it is and due to the fact that p can become... 

This chapter is devoted to the development of perturbation expansions in powers of 1 /c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy-Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit-Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties. The last sections of this chapter are... 

We can classify the Douglas-Kroll expansion in terms of the leading power of c as well as in terms of V. The off-diagonal operator in the zeroth order (free-particle Foldy-Wouthuysen) expansion is Its elimination in the first Douglas-Kroll... 

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. 

According to Eq. (11.93), the decoupled Hamiltonian within the Foldy-Wouthuysen framework is formally given as a series of even terms of well-defined order in 1/c. In most presentations of the Foldy-Wouthuysen transformation the exponential function parametrization Hjj] = exp(W[j]) is applied for each transformation step. However, in the light of the discussion in section 11.4 the specific choice of this parametrization does not matter at all, since one necessarily has to expand Ui into a power series in order to evaluate the Hamiltonian. Consequently, in order to guarantee a most general analysis, the most general parametrization for the Foldy-Wouthuysen transformation should be employed [610]. Thus, li is parametrized as a power series expansion in an odd and antihermitean operator W, , which is of (2/+l)-th order in 1/c, (cf. section 11.4). After n transformation steps, the intermediate, partially transformed Hamiltonian f has the following structure. 

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