Abstract
A general method of the Foldy-Wouthuysen transformation is developed. This method is applicable to relativistic particles with any spin in arbitrarily strong external fields. It can be used when the de Broglie wavelength is much smaller than the characteristic distance. Contrary to previously developed relativistic methods, the present method satisfies the condition of the exact Foldy-Wouthuysen transformation and is well substantiated. The derived relativistic Foldy-Wouthuysen Hamiltonian is expanded in powers of the Planck constant. In this expansion, terms proportional to the zero and first powers are determined exactly in accordance with the above condition and terms proportional to higher powers are not specified. The obtained result agrees with the corresponding formula for the Foldy-Wouthuysen Hamiltonian previously deduced by an iterative relativistic method and proves the validity of results obtained with this formula.
Highlights
The Foldy-Wouthuysen (FW) representation [1] occupies a special place in quantum theory due to its unique properties
The method is general because it is applicable to relativistic particles with any spin in arbitrarily strong external fields
Like other relativistic methods applying an expansion of the FW Hamiltonian in powers of the Planck constant [13,14,15], the proposed method can be used when the de Broglie wavelength is much smaller than the characteristic distance
Summary
The Foldy-Wouthuysen (FW) representation [1] occupies a special place in quantum theory due to its unique properties. A great advantage of the FW representation is the simple form of operators corresponding to classical observables Thanks to these properties, the FW representation provides the best possibility of obtaining a meaningful classical limit of the relativistic quantum mechanics [1, 4]. The passage to the classical limit usually reduces to a replacement of the operators in quantum-mechanical Hamiltonians and equations of motion with the corresponding classical quantities The possibility of such a replacement, explicitly or implicitly used in practically all works devoted to the relativistic FW transformation, was recently rigorously proved in Ref. Some iterative (“step-by-step”) methods of the FW transformation widely used in quantum chemistry [10] possess similar properties They express the exact FW Hamiltonian as a series of relativistic corrections satisfying the Eriksen condition. Like other relativistic methods applying an expansion of the FW Hamiltonian in powers of the Planck constant [13,14,15], the proposed method can be used when the de Broglie wavelength is much smaller than the characteristic distance
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