Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Thomas precession, spin-orbit interaction, and Berry’s phase

Harsh Mathur

Published Dec 9, 1991• 87 citations
Publisher

Abstract

The spin-orbit interaction is shown to arise as a Berry phase term in the adiabatic effective Hamiltonian for the orbital motion of a Dirac electron. This approach makes explicit the intimate connection of the spin-orbit interaction and Thomas precession.
DOI
10.1103/physrevlett.67.3325

Related papers

Spin precession in an adiabatically rotating electric field

Yuri A. Serebrennikov

Due to spin-orbit coupling, the adiabatic perturbation of an electron's orbital motion induced by a revolving external electric field lead to the electron spin-precession. The obtained results describe both transverse and longitudinal dynamics of an effective spin, for conical and figure-8 trajectories of a driving electric field, and elucidate the link between the quantum-mechanical, geometrical, and classical pictures of spin rotation. Within the limit of adiabatic approximation the derived formulas are valid regardless of whether eigenvectors of the total Hamiltonian of the problem are explicitly available or not, and are convenient for approximate calculations. The main expression for the time evolution of the Bloch vector has pure geometric character and is independent of the physical context.

arXiv:cond-mat/0402319•Physical Review B•Aug 30, 2004•9 citations

Derivation of the Spin-Orbit Interaction

Péter Faragó

Many students find it paradoxical that spin-orbit coupling is discussed in nonrelativistic quantum mechanics although it is usually said to be a consequence of relativity. The resolution of this paradox is discussed. The Thomas precession is related to the commutation properties of the matrices used to describe the resultant of two consecutive Lorentz transformations without rotation; the frequency of the Thomas precession is derived on this basis.

American Journal of Physics•Mar 1, 1967•12 citations

On magnetic monopoles, the anomalous g-factor of the electron and the spin-orbit coupling in the Dirac theory

G. Coddens

We discuss the algebra and the interpretation of the anomalous Zeeman effect and the spin-orbit coupling within the Dirac theory. Whereas the algebra for the anomalous Zeeman effect is impeccable and therefore in excellent agreement with experiment, the physical interpretation of that algebra uses images that are based on macroscopic intuition but do not correspond to the meaning of this algebra. The interpretation violates the Lorentz symmetry. We give an alternative intuitive description of the meaning of this effect, which respects the symmetry and is exact. It can be summarized by stating that a magnetic field makes any charged particle spin. We show also that the traditional discussion about magnetic monopoles confuses two issues, viz. the symmetry of the Maxwell equations and the quantization of charge. These two issues define each a different concept of magnetic monopole. They cannot be merged together into a unique all-encompassing issue. We also generalize the minimal substitution for a charged particle, and provide some intuition for the magnetic vector potential. We finally explore the algebra of the spin-orbit coupling, which turns out to be badly wrong. The traditional theory that is claimed to reproduce the Thomas half is based on a number of errors. An error-free application of the Dirac theory cannot account for the Thomas precession, because it only accounts for the instantaneous local boosts, not for the rotational component of the Lorentz transformation. This runs contrary to established beliefs, but Thomas precession can be understood in terms of the Berry phase on a path through the velocity space of the Lorentz group manifold. These results clearly reveal the limitations of the prevailing working philosophy to "shut up and calculate".

arXiv:1604.05160•Apr 1, 2016•2 citations

Origin of the spin–orbit interaction

Gianfranco Spavieri, Masud Mansuripur

We consider a semi-classical model to describe the origin of the spin–orbit interaction in a simple system such as the hydrogen atom. The interaction energy is calculated in the rest-frame of the nucleus, around which an electron, having linear velocity and magnetic dipole-moment travels in a circular orbit. The interaction energy is due to the coupling of the induced electric dipole with the electric field of the nucleus. Assuming the radius of the electron's orbit remains constant during a spin-flip transition, our model predicts that the energy of the system changes by the factor emerging naturally as a consequence of equilibrium and the change of the kinetic energy of the electron. The correct factor for the spin–orbit coupling energy is thus derived without the need to invoke the well-known Thomas precession in the rest-frame of the electron.

arXiv:1506.07239•Physica Scripta•Jun 24, 2015•27 citations

Spin-orbital motion and Thomas precession in the classical and quantum theories

The motion of a magnetic spin particle in electromagnetic fields is considered on the basis of general principles of the classical relativistic theory. Alternative approaches in derivation of the equations of charge motion and spin precession, the problem of noncollinearity of the momentum and velocity of a particle with spin, the origin and the meaning of Thomas precession in dynamics of the spin particle are also considered. The correspondence principle in the spin theory is discussed. 1

•Jan 1, 2001•0 citations

A Method of É. Borel for calculation of the Thomas precession: The geometric phase in relativistic kinematic velocity space and its applications in optics

Grigorii B. Malykin

In 1913, É. Borel (1871–1956) considered the relativistic kinematic rotation of a body that arises in the course of its orbital motion. In order to determine the angular velocity of this rotation, Borel introduced the notion of a kinematic space and applied the Gauss-Bonnet formula for a sphere of a unit radius. However, at that time, his results did not find practical applications and, soon, were forgotten. After the discovery of electron spin by S.A. Goudsmit and G.E. Uhlenbeck in 1925, real interest in this phenomenon was rekindled, and it was considered once again by L.H. Thomas in 1926–1927, being termed the Thomas precession. The Thomas precession manifests itself mostly in optical phenomena.

Optics and Spectroscopy•Feb 1, 2013•0 citations
Due to spin-orbit coupling, the adiabatic perturbation of an electron's orbital motion induced by a revolving external electric field lead to the electron spin-precession. The obtained results describe both transverse and longitudinal dynamics of an effective spin, for conical and figure-8 trajectories of a driving electric field, and elucidate the link between the quantum-mechanical, geometrical, and classical pictures of spin rotation. Within the limit of adiabatic approximation the derived formulas are valid regardless of whether eigenvectors of the total Hamiltonian of the problem are explicitly available or not, and are convenient for approximate calculations. The main expression for the time evolution of the Bloch vector has pure geometric character and is independent of the physical context.
We discuss the algebra and the interpretation of the anomalous Zeeman effect and the spin-orbit coupling within the Dirac theory. Whereas the algebra for the anomalous Zeeman effect is impeccable and therefore in excellent agreement with experiment, the physical interpretation of that algebra uses images that are based on macroscopic intuition but do not correspond to the meaning of this algebra. The interpretation violates the Lorentz symmetry. We give an alternative intuitive description of the meaning of this effect, which respects the symmetry and is exact. It can be summarized by stating that a magnetic field makes any charged particle spin. We show also that the traditional discussion about magnetic monopoles confuses two issues, viz. the symmetry of the Maxwell equations and the quantization of charge. These two issues define each a different concept of magnetic monopole. They cannot be merged together into a unique all-encompassing issue. We also generalize the minimal substitution for a charged particle, and provide some intuition for the magnetic vector potential. We finally explore the algebra of the spin-orbit coupling, which turns out to be badly wrong. The traditional theory that is claimed to reproduce the Thomas half is based on a number of errors. An error-free application of the Dirac theory cannot account for the Thomas precession, because it only accounts for the instantaneous local boosts, not for the rotational component of the Lorentz transformation. This runs contrary to established beliefs, but Thomas precession can be understood in terms of the Berry phase on a path through the velocity space of the Lorentz group manifold. These results clearly reveal the limitations of the prevailing working philosophy to "shut up and calculate".
We consider a semi-classical model to describe the origin of the spin–orbit interaction in a simple system such as the hydrogen atom. The interaction energy is calculated in the rest-frame of the nucleus, around which an electron, having linear velocity and magnetic dipole-moment travels in a circular orbit. The interaction energy is due to the coupling of the induced electric dipole with the electric field of the nucleus. Assuming the radius of the electron's orbit remains constant during a spin-flip transition, our model predicts that the energy of the system changes by the factor emerging naturally as a consequence of equilibrium and the change of the kinetic energy of the electron. The correct factor for the spin–orbit coupling energy is thus derived without the need to invoke the well-known Thomas precession in the rest-frame of the electron.
In 1913, É. Borel (1871–1956) considered the relativistic kinematic rotation of a body that arises in the course of its orbital motion. In order to determine the angular velocity of this rotation, Borel introduced the notion of a kinematic space and applied the Gauss-Bonnet formula for a sphere of a unit radius. However, at that time, his results did not find practical applications and, soon, were forgotten. After the discovery of electron spin by S.A. Goudsmit and G.E. Uhlenbeck in 1925, real interest in this phenomenon was rekindled, and it was considered once again by L.H. Thomas in 1926–1927, being termed the Thomas precession. The Thomas precession manifests itself mostly in optical phenomena.