distributed for no-profit educational & research purposes. Not for commercial use. An isolated static electron in free space is not a fixed particle at rest. It is always oscillating in a SHM in its own electromagnetic inertia field, rest frame even at zero kelvin temperature. This is non-thermal, standing wave, resonance Compton frequency, oscillation along a linear path or, along an elliptical or a circular (clockwise or anti-clockwise) path, corresponding to the electron’s intrinsic magnetic moment (spin up or spin down). An electron with spin behaves like a tiny magnet. Intrinsic spin does not imply that a subatomic particle is spinning like a toytop about its axis. A hypothetical electron without a charge is like the bob of a simple pendulum without a string. INTRINSIC SPIN & ORBITAL ANGULAR MOMENTUM
distributed for no-profit educational & research purposes. Not for commercial use. An isolated static electron in free space is not a fixed particle at rest. It is always oscillating in a SHM in its own electromagnetic inertia field, rest frame even at zero kelvin temperature. This is non-thermal, standing wave, resonance Compton frequency, oscillation along a linear path or, along an elliptical or a circular (clockwise or anti-clockwise) path, corresponding to the electron’s intrinsic magnetic moment (spin up or spin down). An electron with spin behaves like a tiny magnet. Intrinsic spin does not imply that a subatomic particle is spinning like a toytop about its axis. A hypothetical electron without a charge is like the bob of a simple pendulum without a string. INTRINSIC SPIN & ORBITAL ANGULAR MOMENTUM
A solitary electron in free space is not a fixed particle at rest in a given inertial frame of reference. It is always oscillating in a simple harmonic motion (SHM) in its own electromagnetic inertia field, even at zero kelvin temperature. A Newtonian particle exhibits uniform motion in a straight line. A quantum particle like the electron exhibits uniform motion along a sine wave path. This explains the physical concept of the wave-particle nature of the electron. As the mass of a quantum particle increases, the wavelength of the particle decreases as per French physicist Louis de Broglie. In the limiting case the sine wave becomes a straight line. Also, a hypothetical electron without an electric charge is like the bob of a simple pendulum without a string. In an electron-spin qubit, the electron can oscillate either clockwise or anti-clockwise and cannot oscillate both clockwise and anti-clockwise at the same time. It cannot be both spin-up and spin-down at the same time.
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 .
According to the prevailing belief, the spin of the electron or of some other particle is a mysterious internal angular momentum for which no concrete physical picture is available, and for which there is no classical analog. However, on the basis of an old calculation by Belinfante [Physica 6, 887 (1939)], it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitively appealing picture and establishes that neither the spin nor the magnetic moment are ‘‘internal’’—they are not associated with the internal structure of the electron, but rather with the structure of its wave field. Furthermore, a comparison between calculations of angular momentum in the Dirac and electromagnetic fields shows that the spin of the electron is entirely analogous to the angular momentum carried by a classical circularly polarized wave.
Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles, and atomic nuclei. It is wildly believed that spin is a purely quantum mechanical concept and has no classical analogue. In fact, elementary particles are conceived as point objects which have no axis to “spin” around. Therefore, there is no explaining how spin arises at the fundamental level, why particles have the values they do, and what underpins the Pauli Exclusion principle and Bose-Einstein behavior. However, spin is like a vector quantity; it has a definite magnitude, and it has a “direction”, in order to spin should be composite. In this paper we propose a physical explanation for spin of the electron at the sub-particle level, relying on the vortex model of the electron. The electron is described as a superfluid frictionless vortex which has a mass, angular momentum and spin to provide a complete explanation of all properties of the electron: it composite, spinning around its own axis, produces a tiny magnetic fields independent of those from its orbital motions. The classical hydrodynamic laws are used to describe the quantum properties of the electron, such as spin, angular momentum, magnetic momentum and a magnetic dipole. The circulation in the vortex is constant, and the angular momentum of the vortex is conserved and has the same value of Planck constant. The direction of the angular momentum of a spinning electron vortex is along the axis of rotation and determined by the direction of spin. The spin quantum number 1/2 has a fixed value which represents the gap between the circulation rate of the core of the vortex and the boundaries of the vortex. The changeable values +1/2 “spin-up” or -1/2 “spin-down” indicate the direction of the magnetic dipole of the vortex. The relation between spin and Planck constant is discussed and the origin h/4pi angular momentum units are revealed.
The classical equations of motion of a charged point-particle with intrinsic spin under the influence of an external electromagnetic field are restated and compared with the Heisenberg equations of motion derived from the Dirac theory. The partition of angular momentum between particle and field in the classical theory is contrasted to the Dirac theory of electron spin. The analogy between the Dirac equation and the theory of parametric amplification is pointed out.A free spinning point particle moving according to the laws of classical relativistic point-particle mechanics may move along a helix. The sum of the intrinsic spin $\ensuremath{\sigma}$ and the angular momentum of the helical motion in this classical analog of zitterbewegung is an effective spin vector S which is a constant of the motion. Because of this internal motion, the effective mass M of the particle differs from the mass m which is ascribed to it in the equations of motion. Solutions are found in which S is parallel or antiparallel to the momentum, and the sign of M is determined by the helicity. When placed in a uniform electromagnetic field, the particle behaves as if it had a rest mass M and a magnetic moment $\frac{e\ensuremath{\sigma}}{\mathrm{Mc}}$, in addition to any explicit magnetic moment that may be ascribed to it.
The detailed motion of a free Dirac electron is investigated by examining the expectation values of the position r and of r×ṙ in a wave packet. It is shown that the well-known zitterbewegung may be looked upon as a circular motion about the direction of the electron spin, with a radius equal to the Compton wavelength (divided by 2π) of the electron. It is further shown that the intrinsic spin of the electron may be looked upon as the “orbital angular momentum” of this motion. The current produced by the zitterbewegung is seen to give rise to the intrinsic magnetic moment of the electron.