Electron Spin

Electron spin is intrinsic angular momentum carried by the electron. It transforms like angular momentum, couples to magnetic fields, and gives two spin-projection outcomes, but it is not literal rotation of a small extended charged sphere.

Historically, spin was introduced to solve a cluster of spectroscopic problems: anomalous Zeeman splitting, atomic doublets, Pauli’s two-valued quantum number, and the structure of the periodic table. The modern formalism lives in What Spin Is and Is Not and Spin-1/2 Hilbert Space.

Anomalous Zeeman Effect

The normal Zeeman effect could be partly understood by coupling orbital magnetic moments to an external magnetic field. But many atomic spectra showed anomalous splitting patterns that did not fit a spinless orbital picture.

In a simple orbital model, magnetic shifts are tied to angular-momentum projection labels. A magnetic field along $z$ can produce energy shifts of the schematic form

$$\Delta E \propto B m.$$

The observed multiplets, however, required additional structure. Alkali spectra, fine structure, and anomalous Zeeman patterns pointed toward a two-valued internal degree of freedom not reducible to orbital motion alone.

The name “anomalous” is historical. Once electron spin and spin-orbit coupling are included, the patterns are no longer anomalous in the same sense; they are consequences of a richer angular-momentum structure.

Uhlenbeck and Goudsmit Proposal

In 1925, George Uhlenbeck and Samuel Goudsmit proposed that the electron carries an intrinsic angular momentum and magnetic moment. This supplied a physical interpretation for Pauli’s two-valuedness.

The proposal was bold because a literal classical spinning charged sphere led to serious problems. A sufficiently small electron rotating fast enough to produce the required angular momentum would involve impossible classical speeds and incorrect magnetic details. The lesson was not that the electron is a tiny rotating body. The lesson was that a new internal quantum degree of freedom was needed.

Their 1926 Nature paper explicitly refers back to their November 1925 proposal. Thomas’s 1926 analysis supplied the factor of one half missing in an elementary spin–orbit calculation that neglects the relativistic precession of the electron’s instantaneous rest frame. Pauli’s later nonrelativistic spin theory gave a clean two-component wavefunction formalism. Dirac’s relativistic equation then made spin-$1/2$ and the leading electron magnetic moment arise naturally.

Spin Magnetic Moment

Modern nonrelativistic quantum mechanics describes electron spin by operators $\mathbf S$ satisfying the angular momentum algebra. For spin-$1/2$,

$$S^2\lvert s,m_s\rangle = \hbar^2s(s+1)\lvert s,m_s\rangle, \qquad s=\frac12,$$

and

$$S_z\lvert s,m_s\rangle = \hbar m_s\lvert s,m_s\rangle, \qquad m_s=\pm\frac12.$$

The electron spin magnetic moment is conventionally written with $e>0$ as

$$\boldsymbol\mu_s = -g_e\mu_B \frac{\mathbf S}{\hbar}, \qquad \mu_B = \frac{e\hbar}{2m_e}.$$

The minus sign reflects the electron’s negative charge: the electron magnetic moment is antiparallel to its spin angular momentum. Here $g_e$ denotes the positive magnitude; some reference tables instead include the electron’s sign in a signed $g$ factor. Minimal electromagnetic coupling of a point electron in Dirac theory gives $g_e=2$. Precision QED and experiment reveal a small anomalous correction, so the measured value is close to but not exactly $2$.

In a magnetic field, the spin coupling is

$$H_{\mathrm{spin}} = -\boldsymbol\mu_s\cdot\mathbf B.$$

This coupling is the core reason spin appears in magnetic resonance, Zeeman splittings, and Stern–Gerlach-type measurements.

Pauli Theory

Pauli’s 1927 spin theory represented electron spin with two-component wavefunctions and matrices acting on the spin degree of freedom. In a chosen basis, a spin-$1/2$ state has two components:

$$\psi(\mathbf x) = \begin{pmatrix} \psi_\uparrow(\mathbf x)\\ \psi_\downarrow(\mathbf x) \end{pmatrix}.$$

Spin operators are represented using Pauli matrices:

$$S_i = \frac{\hbar}{2}\sigma_i.$$

The probability density is $\lvert\psi_\uparrow(\mathbf x)\rvert^2+\lvert\psi_\downarrow(\mathbf x)\rvert^2$ when spin is not resolved, and its spatial integral is one for a normalized electron state. This formalism made the two-valued degree of freedom operational. It also separated spin from ordinary orbital motion: the electron wavefunction now has both spatial dependence and an internal two-component structure.

The detailed matrix algebra belongs to Pauli Matrices and Spin and Pauli Matrix Conventions. The Hamiltonian reference card is Spin in Magnetic Field Hamiltonian, and the Pauli-equation formula card is Pauli Equation.

Dirac Theory Preview

Dirac’s 1928 relativistic electron equation changed the status of spin. Spin-$1/2$ became part of the representation used by a relativistic electron equation. Relativity alone does not force every particle to have spin-$1/2$; scalar and higher-spin relativistic theories also exist.

The nonrelativistic limit of the Dirac equation yields the Pauli spin coupling with leading $g_e=2$. The same framework also predicted negative-energy structure that later became tied to antiparticles. Full precision treatment of the electron magnetic moment belongs to QED, where radiative corrections explain the small deviation from $2$.

Electron spin began as a solution to spectroscopic anomalies and became a structural part of quantum theory.

Common Misconceptions

• Electron spin is not a tiny ball literally spinning about an axis.
• The original Stern–Gerlach experiment was not a direct free-electron spin measurement.
• Spin-up and spin-down are defined relative to a chosen measurement axis.
• The electron magnetic moment is antiparallel to spin because the electron has negative charge.
• Pauli’s exclusion principle and Pauli matrices are different topics, though both are tied historically to spin.
• The Dirac equation did not make all nonrelativistic spin physics obsolete; it explained why the Pauli theory works in its domain.
• The electron $g$ factor is close to $2$, but precision physics distinguishes Dirac’s leading value from QED corrections.

Exercises

1. What are the allowed values of $m_s$ for an electron, and what are the corresponding $S_z$ eigenvalues?

Solution

For an electron, $s=1/2$ and

$$m_s=\frac12,\quad -\frac12.$$

Since $S_z\lvert s,m_s\rangle=\hbar m_s\lvert s,m_s\rangle$, the eigenvalues are

$$\frac{\hbar}{2}, \qquad -\frac{\hbar}{2}.$$

2. Why is the phrase “spinning electron” historically understandable but physically misleading?

Solution

The phrase reflects the historical attempt to associate the electron’s two-valued degree of freedom and magnetic moment with angular momentum. It is misleading because electron spin is not literal rotation of an extended charged body. It is an intrinsic quantum angular momentum represented by spin operators acting on an internal Hilbert-space degree of freedom.

3. If an electron has $S_z=+\hbar/2$, what is the sign of its spin magnetic moment component $\mu_{s,z}$ using $\boldsymbol\mu_s=-g_e\mu_B\mathbf S/\hbar$?

Solution

Substitute $S_z=+\hbar/2$:

$$\mu_{s,z} = -g_e\mu_B \frac{1}{2}.$$

The component is negative. The spin magnetic moment points opposite the spin direction because the electron charge is negative.

4. What did Dirac’s theory add to Pauli’s nonrelativistic spin theory?

Solution

Pauli’s theory supplied a successful nonrelativistic two-component spin formalism and magnetic coupling. Dirac’s relativistic equation made spin-$1/2$ and the leading $g_e=2$ magnetic moment emerge from a relativistic quantum equation. It also opened the route to antiparticles and later QED corrections.

References

• P. A. M. Dirac, “The Quantum Theory of the Electron,” Proceedings of the Royal Society A 117, 610-624, 1928, DOI: 10.1098/rspa.1928.0023.
• M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
• W. Pauli, “Zur Quantenmechanik des magnetischen Elektrons,” Zeitschrift für Physik 43, 601-623, 1927.
• W. Pauli, “Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren,” Zeitschrift für Physik 31, 765-783, 1925.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• S. S. Schweber, QED and the Men Who Made It, Princeton University Press, 1994.
• L. H. Thomas, “The Motion of the Spinning Electron,” Nature 117, 514, 1926.
• G. E. Uhlenbeck and S. Goudsmit, “Spinning Electrons and the Structure of Spectra,” Nature 117, 264-265, 1926, DOI: 10.1038/117264a0.