Its angular momentum comes from two types of rotation: spin and orbital motion.
If the electron is visualized as a classical rigid body in which the mass and charge have identical distribution and motion that is rotating about an axis with angular momentum L, its magnetic dipole moment μ is given by:
The ratio between the true spin magnetic moment and that predicted by this model is a dimensionless factor ge, known as the electron g-factor:
It is usual to express the magnetic moment in terms of the reduced Planck constant ħ and the Bohr magneton μB:
Classical notions such as the center of charge and mass are, however, hard to make precise for a quantum elementary particle.
A triumph of the quantum electrodynamics theory is the accurate prediction of the electron g-factor.
Here gL is the electron orbital g-factor and μB is the Bohr magneton.
The value of gL is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the classical gyromagnetic ratio.
Here L is the orbital angular momentum, n, ℓ, and m are the principal, azimuthal, and magnetic quantum numbers respectively.
The first to introduce the idea of electron spin was Arthur Compton in his 1921 paper on investigations of ferromagnetic substances with X-rays.
[5]: 145–155 [6] In Compton's article, he wrote: "Perhaps the most natural, and certainly the most generally accepted view of the nature of the elementary magnet, is that the revolution of electrons in orbits within the atom give to the atom as a whole the properties of a tiny permanent magnet.
This pre-1925 period marked the old quantum theory built upon the Bohr-Sommerfeld model of the atom with its classical elliptical electron orbits.
[7] Irving Langmuir had explained in his 1919 paper regarding electrons in their shells, "Rydberg has pointed out that these numbers are obtained from the series
The factor two suggests a fundamental two-fold symmetry for all stable atoms.
configuration was adopted by Edmund Stoner, in October 1924 in his paper 'The Distribution of Electrons Among Atomic Levels' published in the Philosophical Magazine.
Wolfgang Pauli hypothesized that this required a fourth quantum number with a two-valuedness.
The necessity of introducing half-integral spin goes back experimentally to the results of the Stern–Gerlach experiment.
The conclusion is that silver atoms have net intrinsic angular momentum of 1/2.
Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so:
This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function.
Pauli had introduced the 2 × 2 sigma matrices as pure phenomenology — Dirac now had a theoretical argument that implied that spin was somehow the consequence of incorporating relativity into quantum mechanics.
On introducing the external electromagnetic 4-potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form (in natural units ħ = c = 1)
What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles.
This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness.
A further approximation gives the Schrödinger equation as the limit of the Pauli theory.
This also was a great triumph for the new equation, as it traced the mysterious i that appears in it, and the necessity of a complex wave function, back to the geometry of space-time through the Dirac algebra.
It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation.
The entire Dirac spinor represents an irreducible whole, and the components we have just neglected to arrive at the Pauli theory will bring in new phenomena in the relativistic regime – antimatter and the idea of creation and annihilation of particles.
In a general case (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component.
[2] This allows the determination of hyperfine splitting of electron shell energy levels in atoms of protium and deuterium using the measured resonance frequency for several transitions.