It is known that the action of a magnetic field on the electron leads to the precession of its spin. However, if we describe this effect within the Minkowski space, then it turns out that the result of the action is not the precession of spin of electron, but the rotation of the electron with the angular velocity –(e/mc)H, which corresponds to the experimental data (e and m are the charge and mass of the electron, c is the speed of light, H is a magnetic field strength). Bibliogr. 7 references.
It is known that the action of a magnetic field on the electron leads to the precession of its spin. However, if we describe this effect within the Minkowski space, then it turns out that the result of the action is not the precession of spin of electron, but the rotation of the electron with the angular velocity –(e/mc)H, which corresponds to the experimental data (e and m are the charge and mass of the electron, c is the speed of light, H is a magnetic field strength). Bibliogr. 7 references.
First, the experimental foundations for the existence of a new dynamical variable, the spin of particles, are presented. The Pauli equation is then derived in detail, two applications are given and the energy levels of a particle with spin in a constant magnetic field are studied. This is the analysis of Landau levels, which can be performed by using the known results on the spectrum of harmonic oscillators. In this chapter we also study in detail the addition of orbital angular momentum and spin, as well as the spin–orbit interaction with the resulting Thomas precession. Stern-Gerlach experiment and electron spin The hypothesis that the electron has an intrinsic magnetic moment and an angular momentum, in short a spin , was first suggested in Uhlenbeck and Goudsmit (1926). They noticed, even before the discovery of quantum mechanics, that a complete description of spectra was not possible unless a magnetic moment and a mechanical moment were ascribed to the electron, and hence the concept of an electron as a point charge was insufficient. First, let us therefore try to understand how a magnetic moment can be associated to an atomic system. For this purpose, consider for simplicity the Bohr model of an hydrogen atom, where the electron moves along a circular orbit and rotates with orbital angular momentum L .
The behavior of an atom in a nonuniform magnetic field is analyzed, as well as the motion of a classical magnetic dipole (a spinning charged ball) and a rotating charged ring. For the atom it is shown that, while the magnetic field does no work on the electron-orbital contribution to the magnetic moment (the source of translational kinetic energy being the internal energy of the atom), whether or not it does work on the electron-spin contribution to the magnetic moment depends on whether the electron has an intrinsic rotational kinetic energy associated with its spin. A rotational kinetic energy for the electron is shown to be consistent with the Dirac equation. If the electron does have a rotational kinetic energy, the acceleration of a silver atom in a Stern-Gerlach experiment or the emission of a photon from an electron spin flip can be explained without requiring the magnetic field to do work. For a constant magnetic field gradient along the z axis, it is found that the classical objects oscillate in simple harmonic motion along the z axis, the total kinetic energy -- translational plus rotational -- being a constant of the motion. For the charged ball, the change in rotational kinetic energy is associated only with a change in the precession frequency, the rotation rate about the figure axis remaining constant.
In 1925 the concept of electron spin was introduced to explain two experimental observations: the splitting of silver atom beams in an inhomogeneous magnetic field (Stern–Gerlach, 1922), and the fine structure of atomic spectra. The physical picture proposed — an electron spinning on its own axis — was immediately shown by Hendrik Lorentz to require surface velocities exceeding the speed of light. The paper was published anyway. Dirac subsequently rescued the two-valuedness mathematically, whereupon the physical question was declared irrelevant. For one hundred years the field has been unable to state what spin physically is. This paper traces the sequence of observations, inferences, and decisions that produced this situation and identifies two distinct wrong turns. The first wrong turn was publishing a model known to be physically impossible. Lorentz's calculation was correct — at the classical electron radius (2.818 fm), the required surface velocity is 274 times the speed of light. But the classical radius is not the electron's physical rotation radius. The correct rotation radius, derived from the Sagnac closure condition, is 285.5 fm — 101 times larger. At that radius, the rotation velocity is 0.822c, the bare magnetic moment is 0.608 μB, and Lorentz's objection does not arise. Both values are computed explicitly with no free parameters. The second wrong turn was a conflation. The closure geometry of the electron involves two independent numerical facts, both related to the number ½: (1) the least-work oscillation path that closes the Sagnac phase condition is a two-lobe ellipse, giving a winding number of n = ½ in the mass equation; (2) the electron's S³ closure surface presents two poles to the external field, giving a topological factor of 2 in the g-factor. In 1925, both facts were numerically present in the spectroscopic data, but they were collapsed into a single postulate: intrinsic angular momentum ℏ/2. That collapse created a half-integer angular momentum with no classical analogue, forced Dirac to construct the SU(2) spinor formalism to accommodate it, and caused the spinor to appear as a new kind of mathematical object requiring new physics — when it was always a description of closure surface topology. Every experimental result attributed to spin-½ as an intrinsic quantum property — two spots in Stern–Gerlach, the electron magnetic moment, Pauli exclusion, and fine structure — is reproduced by real rotation with two stable orientations, without superluminal motion, without abstract mathematical structure disconnected from physical content, and without the hundred-year admission of ignorance that has accompanied the standard treatment. The mathematical apparatus of quantum mechanics — the Dirac equation, the Pauli matrices, the SU(2) structure — remains valid. What changes is the interpretation. The spinor is not a mysterious new object. It is the natural representation of the two-pole topology of the S³ closure surface. The conflation of 1925 prevented this from being seen. Related work: The Sagnac closure condition and the γcause invariant used in this paper are derived in: D. J. Hallman, "The Sagnac Formula Inverted Reveals Mass, Gravity, and Particle Structure," Zenodo (2026). doi:10.5281/zenodo.20193279
Most students of science and engineering know that every elementary particle, such as electrons, neutrons, photons, neutrinos, etc., has a quantum mechanical property called “spin” which can be measured (perhaps not easily, but at least in principle) and has a quantized value, including zero. The vast majority of these students mentally visualize spin as the angular momentum associated with the elementary particle spinning or rotating about its own axis (like a top or a planetary object). This mental picture, although convenient and comforting, is actually somewhat crude and certainly incomplete. Landau and Lifshitz, in their classic textbook on quantum mechanics [1], wrote “[the spin] property of elementary particles is peculiar to quantum theory. [It] has no classical interpretation... It would be wholly meaningless to imagine the ‘intrinsic’ angular momentum of an elementary particle as being the result of its rotation about its own axis.” The simplistic notion of self-rotation about an axis, shown in Fig. 1.1, can- not explain many features of spin, such as why its magnitude cannot assume continuous values and why it is quantized to certain specific values. It also causes serious problems if taken too literally. As we will see later (Problem 1.2), if we think of an electron as a solid sphere of radius equal to the Lorentz radius e2/(4πǫ0m0c 2) (where e is the electron’s charge, m0 is the mass, c is the speed of light in vacuum, and ǫ0 is the dielectric constant of vacuum), then the velocity on the surface of a rotating electron would have to be many times the velocity of light in vacuum if such a rotation were to generate an angular momentum equal to the electron’s spin. Obviously that would not be permitted by the theory of relativity. Indeed, a deep understanding of quantum mechanics is required to understand how the spin property comes about. Its origin is in relativistic quantum mechanics and really was first appreciated by Paul Andrew Maurice Dirac when he derived the Dirac equation which is the cornerstone of relativistic quantum mechanics. Richard Feynman, noted for his distaste for mysticism in physics, wrote about the notion of “spin”: “It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is down deep in relativistic quantum mechanics.