Leading correction to the relativistic Foldy-Wouthuysen Hamiltonian

For Dirac particles interacting with external fields, we use the exact operator of the Foldy-Wouthuysen transformation obtained by Eriksen and rigorously derive a leading correction in the weak-field approximation to the known relativistic Foldy-Wouthuysen Hamiltonian. For this purpose, we carry out the operator extraction of a square root in the Eriksen operator. The derived correction is important for the scattering of relativistic particles. Since the description of this scattering by a relativistic wave equation of the second order is more convenient, we determine a general connection between relativistic wave equations of the first and second orders. For Dirac particles, the relativistic wave equation of the second order is obtained with a correction similar to that of the Foldy-Wouthuysen Hamiltonian.