The Lowest-Order Foldy-Wouthuysen Transformation

In this section we consider the first step of the Foldy-Wouthuysen transformation, which eliminates the first odd operator of order c. If we employ the idea presented in Eq. (11.57) for this purpose, we may use the antihermitian operator W to remove the odd term Since this odd term cup does 

The expansion coefficients have to satisfy the unitarity conditions (11.60)-(11.66). Transformation of the one-electron operator / now yields 

The condition that one has to impose on W o in order to eliminate the lowest-order odd term occurring in/i then reads [610] 

Due to the very simple structure of [-2]/ which is only a constant and independent of both p and V, inversion of the commutator is straightforward and yields the odd and antihermitean operator [610] 

77) represents infinitely many different unitary parametrizations Ho which all decouple the one-electron Hamiltonian / up to zeroth order in 1/c. The resulting Hamiltonians f, which all differ by the specific choice of expansion coefficients Aq consist of infinitely many terms which can all be assigned a definite order in both 1/c and V. 

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