Non-relativistic approximations of the Dirac Hamitonian

Comparative analysis of direct and “step-by-step” Foldy-Wouthuysen transformation methods

“…(13) with the Hamilton operator expansion in the FW representation in powers of the operators E and O and their products containing sufficiently large powers obtained by the Eriksen method. Such an expansion was found in [16]. But the result there, obtained using symbolic computer computations, is represented in a form very inconvenient for this comparison.…”

“…(4) as the initial and formal expression for the Hamiltonian in the FW representation in terms of the operators E and O. The transformed Hamiltonian was represented in precisely this form in [10,11,16,23].…”

“…Because all these methods are approximate, we must determine their applicability boundaries. Obviously, the simplest, most reliable way to do this is to compare relativistic Hamiltonians in the FW representation obtained by methods of the step-by-step type with the exact power series given in [16]. This problem is extremely important, in particular, because the terms proportional to the second derivatives of the field potentials and to the squared field strengths are checked.…”

“…Reducing the Hamiltonian operator found in [16] by writing it via multiple commutators leads to the expression…”

“…These two terms are respectively of the zeroth and first orders in /S 0 . Subsequent terms in (16) and (17) (13) and (17) already appear in the result of the first transformation step, and they must be taken into account in the framework of the weak-field approximation (E ≪ m) when considering terms of the order ( /S 0 ) 2 . They arise as a result of transforming the first double commutator in Eq.…”