General properties of the Foldy-Wouthuysen transformation which is widely used in quantum mechanics and quantum chemistry are considered. Merits and demerits of the original Foldy-Wouthuysen transformation method are analyzed. While this method does not satisfy the Eriksen condition of the Foldy-Wouthuysen transformation, it can be corrected with the use of the Baker-Campbell-Hausdorff formula. We show a possibility of such a correction and propose an appropriate algorithm of calculations. An applicability of the corrected Foldy-Wouthuysen method is restricted by the condition of convergence of a series of relativistic corrections.
The exact exponential Foldy-Wouthuysen transformation operator applicable for a particle with an arbitrary spin is derived. It can be successfully utilized for verifying any Foldy-Wouthuysen transformation method based on the exponential operator. When a verified method is relativistic, the relativistic exponential operator should be expanded in the semirelativistic power series. The obtained exponential operator can be also used for a derivation of the Foldy-Wouthuysen Hamiltonian and its comparison with Hamiltonians found by other methods. This procedure makes it possible to check the validity of any other method of the Foldy-Wouthuysen transformation.
The exact exponential Foldy-Wouthuysen transformation operator applicable for a particle with an arbitrary spin is derived. It can be successfully utilized for verifying any Foldy-Wouthuysen transformation method based on the exponential operator. When a verified method is relativistic, the relativistic exponential operator should be expanded in the semirelativistic power series. The use of the obtained exponential operator for a derivation of the Foldy-Wouthuysen Hamiltonian makes it possible to check the validity of any other method of the Foldy-Wouthuysen transformation, while this possibility may need a greater computational effort.