Foldy-Wouthuysen Transformation of the Breit Equation

The derivation of the Breit-Pauli Hamiltonian is tedious. It is nowadays customary to follow the Foldy-Wouthuysen approach first given by Chraplyvy [679-681], which has, for instance, been sketched by Harriman [59]. Still, many presentations of this derivation lack significant details. In the spirit of this book, we shall give an explicit derivation which is as detailed and compact as possible. A review of the same expression derived differently was provided by Bethe in 1933 [72]. Compared to the DKH treatment of the two-electron term to lowest order as described in section 12.4.2 we now only consider the lowest-order terms in 1/c. We transform the Breit operator in Eq. (8.19) by the free-particle Foldy-Wouthuysen transformation. 

For the sake of simplicity, we have chosen flo,o = o,l = 1 in the expansion of the unitary matrix above with the odd and antihermitean parameters, W[q] (1) and W[o](2), defined by Eq. (11.80). Its structure is most easily expressed in terms of the exponential parametrization of Eq. (11.49), 

These W[q] operators are to be understood to operate only on the corresponding coordinates. Hence, 

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