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A general method of the Foldy-Wouthuysen transformation is developed. Thismethod is applicable to relativistic particles with any spin in arbitrarilystrong external fields. It can be used when the de Broglie wavelength is muchsmaller than the characteristic distance. Contrary to previously developedrelativistic methods, the present method satisfies the condition of the exactFoldy-Wouthuysen transformation and is well substantiated. The derivedrelativistic Foldy-Wouthuysen Hamiltonian is expanded in powers of the Planckconstant. In this expansion, terms proportional to the zero and first powersare determined exactly in accordance with the above condition and termsproportional to higher powers are not specified. The obtained result agreeswith the corresponding formula for the Foldy-Wouthuysen Hamiltonian previouslydeduced by an iterative relativistic method and proves the validity of resultsobtained with this formula.

General method of the relativistic Foldy-Wouthuysen transformation and proof of validity of the Foldy-Wouthuysen Hamiltonian