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arXiv:0910.5155

Foldy-Wouthuysen Transformation and Semiclassical Transition for Relativistic Quantum Mechanics

Alexander J. Silenko

Published Oct 27, 2009• 0 citations
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Abstract

It is shown that the Foldy-Wouthuysen transformation for relativistic particles in strong external fields provides the possibility of obtaining a meaningful classical limit of the relativistic quantum mechanics. The full agreement between quantum and classical theories is proved. The coincidence of the semiclassical equations of motion of particles and their spins with the corresponding classical equations is established. The Niels Bohr's correspondence principle is valid not only in the limit of large spin quantum numbers but also for particles with any spin as well as for spinless particles.

DOI
10.48550/arxiv.0910.5155

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It is shown that the Foldy-Wouthuysen transformation for relativistic particles in strong external fields provides the possibility of obtaining a meaningful classical limit of the relativistic quantum mechanics. The full agreement between quantum and classical theories is proved. The coincidence of the semiclassical equations of motion of particles and their spins with the corresponding classical equations is established. The Niels Bohr's correspondence principle is valid not only in the limit of large spin quantum numbers but also for particles with any spin as well as for spinless particles.
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The classical dynamics for a charged spin particle is governed by the Lorentz force equation for orbital motion and by the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation for spin precession. In static and homogeneous electromagnetic fields, it has been shown that the Foldy-Wouthuysen (FW) transform of the Dirac-Pauli Hamiltonian, which describes the relativistic quantum theory for a spin-1/21/21/2 particle, is consistent with the classical Hamiltonian (with both the orbital and spin parts) up to the order of 1/m141/{m}^{14}1/m14 (mmm is the particle's mass) in the low-energy and weak-field limit. In this paper, we extend this correspondence to the case of inhomogeneous fields. Regardless of the field gradient (e.g. Stern-Gerlach) force, the T-BMT equation is unaltered and thus the classical Hamiltonian remains the same, but subtleties arise and need to be clarified. For the relativistic quantum theory, we apply Eriksen's method to obtain the exact FW transformations for the two special cases, which in conjunction strongly suggest that, in the weak-field limit, the FW transformed Dirac-Pauli Hamiltonian (except for the Darwin term) is in agreement with the classical Hamiltonian in a manner that classical variables correspond to quantum operators via a specific Weyl ordering. Meanwhile, the Darwin term is shown to have no classical correspondence.
It has long been speculated that the Dirac or, more generally, the Dirac-Pauli spinor in the Foldy-Wouthuysen (FW) representation should behave like a classical relativistic spinor in the low-energy limit when the probability of particle-antiparticle pair creation and annihilation is negligible. In the weak-field limit of static and homogeneous electromagnetic fields, by applying the method of direct perturbation theory inductively on the orders of 1/c1/c1/c in the power series, we rigorously prove that it is indeed the case: the FW transformation of the Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.