Electron Spin and the Spin Magnetic Moment

Fire a thin beam of silver atoms through a lopsided magnet — one pole a knife edge, the other a flat face — and the plate answers with two crisp spots, not the vertical smear a classical bar magnet would paint. That two-ness is the electron announcing a property with no analogue in orbital motion: an intrinsic angular momentum of fixed magnitude, spin. The algebra behind those spots — $\hat S^2,\hat S_z$ , their half-integer ladder, the Pauli matrices with $\hat{\mathbf S}=\tfrac\hbar2\boldsymbol\sigma$ — is built in aq-qm-spin-angular-momentum and aq-qm-spin-half-pauli-matrices; the moment that lets a field grab the spin, $\boldsymbol\mu_s=-g_s(\mu_B/\hbar)\hat{\mathbf S}$ , and the Bohr magneton $\mu_B=e\hbar/2m_e$ that scales it, are the business of aq-qm-magnetic-moment-gyromagnetic-ratio. Quantum mechanics builds those operators; this article photographs them — the discovery experiments, the instrument numbers, the resonance dials that convert $g_s\approx2$ into gigahertz. The one surprise that separates spin from every classical picture is the factor of two. A circulating charge is a current loop with $g=1$ , and a rigid spinning charged ball also gives $g=1$ — yet the electron's spin is twice as magnetic per unit of angular momentum, $g_s\approx2.00232$ , which is exactly why it cannot be anything spinning in space. Dirac's relativistic equation delivers the $2$ automatically, and quantum electrodynamics computes the trailing $0.00232$ to twelve digits. Downstream, this doubled moment coupled to the atom's own internal field is what splits the sodium D line into a doublet, drives the anomalous Zeeman effect, and underwrites electron spin resonance — the spectroscopic fingerprints this band of the wiki is about.

Key takeaways

The Stern–Gerlach beam splits into exactly two — an even count no orbital angular momentum ( $2\ell+1$ , always odd) can produce — forcing $s=\tfrac12$ , projections $m_s=\pm\tfrac12$ ; the vector magnitude $|\mathbf S|=\tfrac{\sqrt3}{2}\hbar\approx0.866\,\hbar$ exceeds its projection $\tfrac\hbar2$ .
The spin magnetic moment is $\boldsymbol\mu_s=-g_s(\mu_B/\hbar)\hat{\mathbf S}$ with $\mu_B=e\hbar/2m_e=9.274\times10^{-24}$ J/T $=57.88\ \mu$ eV/T; the minus sign is the electron's charge.
The g-factor is anomalously near two: Dirac gives $g_s=2$ exactly, QED refines it to $g_s=2.00231930436$ , the anomaly $a_e=(g_s-2)/2\approx1.16\times10^{-3}$ led by Schwinger's $\alpha/2\pi$ .
Read as frequency, $g_s$ is a dial: an electron spin flips at $g_s\mu_B B/h=28.02$ GHz/T, so an X-band spectrometer at $0.35$ T sits at $9.81$ GHz.
Key idea: the electron carries an intrinsic $\hat{\mathbf S}$ with $s=\tfrac12$ and a moment $\boldsymbol\mu_s=-g_s(\mu_B/\hbar)\hat{\mathbf S}$ ; the g-factor $g_s\approx2$ — not 1 — is a relativistic/QED effect that gives spin twice the magnetic response per unit angular momentum of orbital motion, and it is what makes spin spectroscopically loud.

The core insight

The electron possesses an intrinsic angular momentum $\hat{\mathbf S}$ with $s=\tfrac12$ and a magnetic moment $\boldsymbol\mu_s=-g_s(\mu_B/\hbar)\hat{\mathbf S}$ ; the g-factor $g_s\approx2$ (not 1) is a relativistic and QED effect, so spin is twice as magnetic per unit of angular momentum as an orbit — the origin of the Stern–Gerlach spots and of every spin-driven splitting downstream.

Where this lives today

The spin moment is working infrastructure across three frequency bands. Electron spin resonance (ESR/EPR) spectrometers — the $9.81$ GHz X-band machine of the worked example — read the $g$ -factor and its shifts to diagnose radicals, transition-metal centres, and defects in semiconductors, since a $g$ pulled off $2.0023$ fingerprints the electron's environment. A single electron in a Penning trap yields $g_s$ to better than a part in $10^{12}$ — at once the sharpest test of quantum electrodynamics and one of the best determinations of $\alpha$ . And at radio frequencies the same physics with a nuclear moment is magnetic resonance imaging: hydrogen nuclei precessing at $42.58$ MHz/T, tipped and read as the resonance derivation describes. Every one descends from the two spots on Gerlach's plate.

The physical picture

Spin's visibility is a single sentence: a uniform field only torques a magnetic moment, but an inhomogeneous field exerts a net force, $F_z=\mu_z\,\partial B_z/\partial z$ , so a gradient sorts atoms by $\mu_z$ . Classically $\mu_z$ runs continuously from $-\mu$ to $+\mu$ and the beam should smear into a band. The silver atom's lone valence electron offers instead exactly two values of $S_z$ — two forces, two spots. The measurement does not read a pre-existing tilt; it forces one of two answers to "up or down along this axis?", and the axis is the experimenter's to choose.

Figure

The Stern–Gerlach experiment: a silver beam from a 1230 K oven crosses a knife-edge magnet (dB/dz ≈ 1400 T/m, length L = 3.5 cm) and lands as TWO spots split by ≈ 0.31 mm — the classical smear it should have made is ghosted behind

Two features close the picture. First, because the electron's charge is negative the moment points opposite the spin, so a spin-up electron is a moment-down magnet pushed toward weaker field; the spot labels are a sign convention, the splitting is not. Second — the fact that makes spin loud rather than a curiosity — the moment is twice as strong as its angular momentum would classically allow. That doubling is a pure number, $g_s$ , and reading it as a resonance frequency turns it into the working dial of magnetic spectroscopy: one tesla flips an electron spin at $28.02$ GHz.

Figure

The punchline dial: an electron spin flips at ν = g_s μ_B B/h, a straight line of slope 28.02 GHz/T (blue). An X-band spectrometer working at B = 0.35 T reads ν = 9.81 GHz (marked). A hypothetical g = 1 orbital moment would give only half the slope, 14.0 GHz/T (amber, dashed)

The core equations

Spin obeys the same angular-momentum algebra as orbital motion, so its spectrum has the same shape — but with one quantum number frozen. The eigenvalue relations below are recalled from aq-qm-spin-angular-momentum, where the ladder that fixes them is derived; here they are the starting facts.

\hat S^2\,|s,m_s\rangle=s(s+1)\hbar^2\,|s,m_s\rangle,\qquad \hat S_z\,|s,m_s\rangle=m_s\hbar\,|s,m_s\rangle \qquad (1)

where: $s$ — spin quantum number, fixed at $\tfrac12$ (a species label, never excited); $m_s=\pm\tfrac12$ — projection; $\hbar$ — reduced Planck constant. That $s$ is frozen while only $m_s$ is dynamical makes the electron a permanent two-level system. The magnitude of the spin vector exceeds its largest projection — the signature of quantized angular momentum, that the arrow tilts on a cone and never points flat along the axis.

|\mathbf S|=\sqrt{s(s+1)}\,\hbar=\frac{\sqrt3}{2}\hbar\approx0.866\,\hbar,\qquad S_z=\pm\frac\hbar2 \qquad (2)

where: $|\mathbf S|$ — length of the spin vector, $\approx0.866\,\hbar$ ; $S_z$ — its only sharp component, $\pm\hbar/2$ . A common slip is to call the magnitude $\tfrac12\hbar$ ; it is $\tfrac{\sqrt3}{2}\hbar$ , half again as large. Attached to the spin is a magnetic dipole moment — the handle every field grabs. Its form is the gyromagnetic relation of aq-qm-magnetic-moment-gyromagnetic-ratio, specialized to spin with the anomalous factor $g_s$ .

\boldsymbol\mu_s=-g_s\,\frac{\mu_B}{\hbar}\,\hat{\mathbf S},\qquad \mu_{s,z}=-g_s\,\mu_B\,m_s=\mp\frac{g_s}{2}\mu_B \qquad (3)

where: $\boldsymbol\mu_s$ — spin magnetic moment; $g_s\approx2.00232$ — spin g-factor; $\mu_B$ — Bohr magneton; the minus sign encodes $q=-e$ , so $\boldsymbol\mu_s$ is antiparallel to $\hat{\mathbf S}$ . Numerically $|\mu_{s,z}|=\tfrac{g_s}{2}\mu_B\approx1.001\,\mu_B$ — half a unit of angular momentum, one full magneton of magnetism. The Bohr magneton is the quantum of atomic magnetism, one unit of orbital angular momentum converted at the classical rate $e/2m_e$ ; its derivation is owned by aq-qm-magnetic-moment-gyromagnetic-ratio.

\mu_B=\frac{e\hbar}{2m_e}=9.274\times10^{-24}\ \mathrm{J/T}=57.88\ \mu\mathrm{eV/T} \qquad (4)

where: $e$ — elementary charge; $m_e$ — electron mass. As a frequency this is $\mu_B/h=13.996$ GHz/T; as a temperature $\mu_B/k_B=0.672$ K/T — the conversions that turn a moment into a dial. The g-factor refuses to be one. Dirac's equation gives exactly two for a point spin- $\tfrac12$ particle; the vacuum then dresses the electron in virtual photons, shifting it by the anomaly.

g_s=2\left(1+\frac{\alpha}{2\pi}+\cdots\right)=2.00231930436,\qquad a_e\equiv\frac{g_s-2}{2}\approx1.16\times10^{-3} \qquad (5)

where: $\alpha\approx1/137.036$ — fine-structure constant; $a_e$ — electron anomaly, led by Schwinger's one-loop $\alpha/2\pi\approx1.16\times10^{-3}$ . Measured and computed values agree to about a part in $10^{12}$ , the most precise confrontation of theory and experiment in physics. Finally, put the moment in a field: the energy is the classical dipole energy promoted to an operator, the two states split, and the gap read as a photon is the spin-flip resonance.

\hat H=-\boldsymbol\mu_s\cdot\mathbf B=g_s\,\mu_B\,m_s\,B,\qquad h\nu=g_s\,\mu_B\,B \qquad (6)

where: $\mathbf B=B\hat z$ — applied field; the gap is $g_s\mu_B B$ , its photon frequency $\nu=g_s\mu_B B/h=28.02\ B$ GHz ( $B$ in tesla). This one line is electron spin resonance.

Derivation AStern–Gerlach: turning two values of the moment into two spots on a plate

The goal is to show, with every number, that a beam of neutral atoms carrying a two-valued magnetic moment splits into two resolvable spots in a field gradient — the experimental birth certificate of spin. We compute the force a gradient exerts on a dipole, insert the quantized moment so it takes only two values, then run the ballistics of the Stern–Gerlach geometry (silver from a $1230$ K oven, a knife-edge magnet of length $L=3.5$ cm and gradient $\partial B_z/\partial z=1400$ T/m) to an actual deflection in millimetres. This is owned here as instrument physics; the spin algebra it rests on is recalled from aq-qm-spin-angular-momentum.

Step 1. A dipole in a field has energy $U=-\boldsymbol\mu\cdot\mathbf B$ ; with $\mathbf B$ predominantly along $z$ , only $\mu_z$ matters and the force is minus the gradient,

F_z=-\frac{\partial U}{\partial z}=\mu_z\,\frac{\partial B_z}{\partial z}.

Why a gradient and not the field itself: a uniform $\mathbf B$ gives a position-independent $U$ and therefore zero net force — only a spatially varying field can deflect a neutral atom. This is why one pole is a knife edge.

Step 2. Insert $\mu_z=-g_s\mu_B m_s$ with $m_s=\pm\tfrac12$ ; since $g_s/2\approx1$ the projection is $\mu_z=\mp\mu_B$ , so

F_z=\mp\mu_B\,\frac{\partial B_z}{\partial z}=\mp(9.274\times10^{-24})(1400)=\mp1.298\times10^{-20}\ \mathrm{N}.

Why exactly two forces: $m_s$ takes only two values, so an orbital $\ell$ would give the odd count $2\ell+1$ ; observing two demands $2s+1=2$ , i.e. $s=\tfrac12$ — the plate reads the orientation count directly.

Step 3. The atoms leave the $1230$ K oven at the flux-weighted speed $v=\sqrt{3k_BT/m}$ ; with $m=107.87\,u$ this is $v=533$ m/s, so the flight time is $t=L/v=6.6\times10^{-5}$ s and each beam accelerates sideways by $a=F_z/m=7.24\times10^{4}$ m/s². Why $\sqrt{3k_BT/m}$ and not the r.m.s. or peak speed: a beam is flux-weighted (faster atoms are over-represented in the emergent stream), which shifts the characteristic speed up to this value.

Step 4. The one-sided deflection at the magnet exit is $z=\tfrac12 a t^2=\tfrac12(7.24\times10^{4})(6.6\times10^{-5})^2=1.6\times10^{-4}$ m, so

z_{\text{one-sided}}=0.156\ \mathrm{mm},\qquad \text{full split }2z=0.31\ \mathrm{mm}.

Why this settled the matter in 1922: a third of a millimetre is comfortably resolvable, and it is a clean doubling with nothing in the middle — the continuous classical band never appears. *Verified in verify_aq-optics-electron-spin-magnetic-moment.py.*

Go deeper · Derivation B · 1 minThe spin-flip resonance: from precession to the ESR dial

Derivation BThe spin-flip resonance: from precession to the ESR dial

The goal is the resonance condition every electron-spin-resonance spectrometer is built on, $h\nu=g_s\mu_B B$ , derived semiclassically from the torque on the spin moment and read as a frequency-per-tesla dial. We write the moment's equation of motion in a uniform field, recognise it as precession at the Larmor rate, and identify the photon that flips the spin. The exact two-level dynamics — the driven Rabi problem behind a real pulse — is owned by aq-qm-spin-half-pauli-matrices and aq-qm-magnetic-moment-gyromagnetic-ratio; here we turn the resonance frequency into an instrument reading.

Step 1. In a uniform field the moment feels a torque $\boldsymbol\tau=\boldsymbol\mu_s\times\mathbf B$ , and because $\boldsymbol\mu_s$ is chained to $\hat{\mathbf S}$ the expectation value obeys the gyroscopic equation

\frac{d\langle\hat{\mathbf S}\rangle}{dt}=-g_s\frac{\mu_B}{\hbar}\,\mathbf B\times\langle\hat{\mathbf S}\rangle.

Why precession rather than alignment: a torque on an angular momentum turns it sideways, changing direction not magnitude — absent dissipation the spin sweeps a cone about $\mathbf B$ forever, never settling like a compass needle.

Step 2. The transverse components rotate rigidly at the Larmor frequency $\omega_L=g_s\mu_B B/\hbar$ while the component along the field is frozen, $d\langle S_z\rangle/dt=0$ . Why $\langle S_z\rangle$ is conserved: the energy $g_s\mu_B m_s B$ is a constant of the motion, so the cone angle is fixed — only the phase of the transverse spin moves.

Step 3. The two spin states are separated in energy by $\Delta E=g_s\mu_B B$ — exactly $\hbar\omega_L$ — so a photon resonant with the precession flips the spin:

h\nu=g_s\mu_B B\ \Longrightarrow\ \nu=\frac{g_s\mu_B}{h}\,B=28.02\ B\ \mathrm{GHz}\ \ (B\ \text{in T}).

Why the resonance and precession frequencies coincide for spin- $\tfrac12$ : with only two levels the Zeeman gap and the classical precession are the same number, so the resonance is read directly off the field.

Step 4. At an X-band working field $B=0.35$ T the dial reads $\nu=28.02\times0.35=9.81$ GHz — microwave territory, the standard ESR band. The same law with an orbital $g=1$ would read half as fast, $14.0\ B$ GHz. Verified in verify_aq-optics-electron-spin-magnetic-moment.py.

Go deeper · Derivation C · 2 minCounting to s = 1/2 and being forced to g ≈ 2

Derivation CCounting to s = 1/2 and being forced to g ≈ 2

The goal is the inference Uhlenbeck and Goudsmit actually made: that beam and spectroscopic evidence together fix both the size of the spin ( $s=\tfrac12$ ) and the anomalous strength of its moment ( $g_s\approx2$ ), neither of which classical physics volunteers. We read three clues — the Stern–Gerlach count, the alkali doublets, and the size of the fine-structure and Zeeman splittings — and show they reconcile only with a half-integer angular momentum carrying a double-strength moment. This reconstructs the 1925 argument; the operator that later made it rigorous, $\boldsymbol\mu=g\gamma\hat{\mathbf J}$ , is owned by aq-qm-magnetic-moment-gyromagnetic-ratio.

Step 1. The beam splits in two. An orbital angular momentum offers $2\ell+1$ orientations, always odd, so a two-fold split cannot be orbital; it needs $2s+1=2$ , hence $s=\tfrac12$ with $m_s=\pm\tfrac12$ . Why this is decisive: no wavefunction on a sphere supplies a half-integer, so the two-ness is not a small orbital effect misread — it is a genuinely new degree of freedom.

Step 2. Spectral lines of the alkalis come in close doublets even though a single valence electron in a central field should emit single lines. A second two-valued label on the electron doubles the state count exactly as observed. Why it must be the same label: the doublets and the two spots both demand a two-valued internal property, and economy — plus the fine-structure ordering — identifies them as one and the same spin.

Step 3. The observed anomalous Zeeman patterns and fine-structure splittings come out a factor of two too small if the spin moment obeys the classical $g=1$ ; matching them requires

g_s\approx2,\qquad \boldsymbol\mu_s=-g_s\frac{\mu_B}{\hbar}\hat{\mathbf S}.

Why $g=1$ fails: both a current loop and a rigid spinning ball give $g=1$ , so the doubling cannot come from any picture of moving charge — the tell that spin is intrinsic. The residual factor-of-two in the fine structure itself is repaired separately by Thomas's relativistic precession, a story for the fine-structure serial.

Step 4. Assemble: an intrinsic $s=\tfrac12$ with a near-double moment accounts for the spots, the doublets, and the splittings at once. Dirac (1928) later derived $g_s=2$ from relativity, and QED the trailing anomaly — but the data forced the inference first. Verified in verify_aq-optics-electron-spin-magnetic-moment.py.

Limiting cases

Case	Result	Meaning
classical spinning ball	$g=1$	a rigid charged sphere gives moment-to-angular-momentum equal to the orbital case — off by the crucial factor of two, proof spin is not classical
Dirac (tree level)	$g_s=2$ exactly	the relativistic equation for a point spin- $\tfrac12$ particle yields two with no adjustable parameter
QED-corrected	$g_s=2.00231930436$	Schwinger's $\alpha/2\pi$ leads the anomaly $a_e\approx1.16\times10^{-3}$ ; theory and experiment meet at $\sim10^{-12}$
$g_s\to1$ (hypothetical)	ESR slope halves to $14.0$ GHz/T	reading the g-factor as a frequency shows the doubling directly — the spin line is twice as steep as an orbital one
$\mu_B B\ll k_BT$	$\mu_B(1\,\mathrm T)/k_B(300\,\mathrm K)=2.24\times10^{-3}$	at room temperature a tesla barely polarizes a spin ensemble — why paramagnetism is weak and spin devices like cold
large $s$	$2s+1\to$ continuum	many-rung ladders blur toward the classical fixed-length vector; the two-spot strangeness lives only at $s=\tfrac12$

Common trapThe electron is a tiny ball of charge, spinning

The name is a fossil and the picture fails twice. Kinematically: give the electron the largest radius any scattering experiment allows and demand its surface store $\tfrac12\hbar$ — the equator would have to move well over a hundred times the speed of light, and the electron is pointlike as far as we can tell. Structurally: the spinning-ball model predicts $g=1$ , but the electron's $g_s\approx2$ , a factor only the Dirac equation explains — so even the one thing the cartoon gets right (a charge with angular momentum is a magnet) it gets wrong by two. Two more habits to drop: $s=\tfrac12$ is fixed while only $m_s=\pm\tfrac12$ is dynamical, and the magnitude is $\tfrac{\sqrt3}{2}\hbar\approx0.866\,\hbar$ , not $\tfrac12\hbar$ .

Worked examples

1 · The X-band ESR dial (a real instrument number)

A benchtop electron-spin-resonance spectrometer operates in the microwave X band. Its electromagnet is set to $B=0.350$ T. At what frequency does a free-electron spin ( $g_s=2.00232$ ) come into resonance, and how does that compare with a proton at the same field?

Step 1. The resonance condition is $h\nu=g_s\mu_B B$ , so $\nu=g_s\mu_B B/h$ . Using $\mu_B/h=13.996$ GHz/T, the spin slope is $g_s\times13.996=28.02$ GHz/T.

Step 2. Multiply by the field: $\nu=28.02\times0.350=\mathbf{9.81\ GHz}$ — squarely in the X band ( $8$ – $12$ GHz), which is exactly why X-band hardware and this field are paired in commercial ESR.

Step 3. A proton has $g_p\approx5.586$ but a nuclear magneton $1836$ times smaller, so its dial is $42.58$ MHz/T; at $0.350$ T it answers at $14.9$ MHz — a factor $\sim660$ below the electron. The electron flips at 9.81 GHz (microwave), the proton at 14.9 MHz (radio) — mass in the denominator is why ESR lives in microwaves and NMR in radio.

2 · The Stern–Gerlach deflection, end to end

Silver atoms ( $m=107.87\,u$ ) leave a $1230$ K oven, are collimated, and cross a knife-edge magnet with gradient $\partial B_z/\partial z=1400$ T/m over a length $L=3.5$ cm. Take the spin projection $|\mu_z|=\mu_B$ . Find the beam speed, the deflecting force, and the spot separation.

Step 1. Beam speed: $v=\sqrt{3k_BT/m}=\sqrt{3(1.381\times10^{-23})(1230)/(107.87\times1.661\times10^{-27})}=\mathbf{533\ m/s}$ .

Step 2. Force: $F_z=\mu_B\,\partial B_z/\partial z=(9.274\times10^{-24})(1400)=\mathbf{1.30\times10^{-20}\ N}$ , one sign for each $m_s$ .

Step 3. Ballistics: acceleration $a=F_z/m=7.24\times10^{4}$ m/s²; flight time $t=L/v=6.57\times10^{-5}$ s; one-sided deflection $z=\tfrac12 a t^2=\mathbf{0.156\ mm}$ .

Step 4. The two beams deflect oppositely, so the plate shows a split $2z=\mathbf{0.31\ mm}$ with nothing between. A third of a millimetre, cleanly doubled — the 1922 result that put a two-valued angular momentum into physics.

Physics of the portrait

Spin was proposed by two graduate students who then tried to unpropose it — and were saved only because their supervisor had already mailed the paper. The concept sits under a person because the inference, not any single equation, is the achievement: reading a factor of two out of spectra and a beam.

Origins · 2 minGeorge Uhlenbeck and how this was discovered

George Uhlenbeck

1900–1988 · Leiden, then Ann Arbor

In the autumn of 1925 George Uhlenbeck and Samuel Goudsmit, both students of Paul Ehrenfest at Leiden, noticed that the alkali doublets and the anomalous Zeeman patterns all fell into place if the electron itself carried an angular momentum of $\tfrac12\hbar$ with a moment of double the classical strength. Uhlenbeck, the more theoretically minded of the pair, worried about the mechanics and took the idea to Hendrik Lorentz, who returned a courteous calculation showing that a charged sphere storing that much angular momentum would have to spin with its surface faster than light. Alarmed, the two asked Ehrenfest to hold the manuscript back; he told them he had already sent it off, adding that they were "young enough to be able to afford a stupidity." The stupidity, it turned out, was only the mechanical picture — a spinning ball — and not the physics. Within months Llewellyn Thomas supplied the relativistic factor of one-half that reconciled the doublet widths, and by 1927 Wolfgang Pauli had given spin its two-component matrix formalism; a year later Dirac derived $g_s=2$ from relativity outright. Uhlenbeck's later career turned to statistical mechanics — the Ornstein–Uhlenbeck process of Brownian motion is his, developed with Leonard Ornstein — and he spent much of it at the University of Michigan in Ann Arbor. But the spin gamble remains the textbook case of trusting the data over the picture: the electron does not spin, and it has spin.

History

1915

— Einstein and de Haas measure the rotation induced by remagnetizing an iron rod, tying magnetism to mechanical angular momentum

1922

— Stern and Gerlach split a silver beam into two spots: space quantization, photographed

1925

— Uhlenbeck and Goudsmit propose intrinsic electron spin with s = 1/2 and a double-strength moment

1926

— Thomas's relativistic precession supplies the missing factor of one-half in the fine structure

1927

— Phipps and Taylor repeat Stern–Gerlach with hydrogen, confirming the electron alone carries the spin

1928

— Dirac's relativistic equation yields g = 2 for a point spin-1/2 particle

1944

— Zavoisky observes electron spin resonance directly, the microwave dial made real

1947

— Kusch and Foley measure the electron g-factor slightly above two, opening the QED anomaly

Going further

Texts. Foot, Atomic Physics, chs. 5–6 (spin, Stern–Gerlach, magnetic moments) · Demtröder, Atoms, Molecules and Photons, ch. 5 · Bransden & Joachain, Physics of Atoms and Molecules, ch. 1 (Stern–Gerlach and spin) · Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §4.4 (spin formalism) · CODATA 2018 recommended values (Bohr magneton, electron g-factor).

Provenance. The spin algebra $\hat S^2,\hat S_z$ and the Pauli representation $\hat{\mathbf S}=\tfrac\hbar2\boldsymbol\sigma$ are recalled by name from aq-qm-spin-angular-momentum and aq-qm-spin-half-pauli-matrices; the moment relation $\boldsymbol\mu=g\gamma\hat{\mathbf J}$ , the Bohr magneton, $g_s\approx2$ , and the Landé $g_J$ used to connect to the Zeeman effect are recalled from aq-qm-magnetic-moment-gyromagnetic-ratio (not re-derived here). A verifier should confirm: (1) $\mu_B=e\hbar/2m_e=9.274\times10^{-24}$ J/T and $\mu_B/h=13.996$ GHz/T; (2) $g_s=2.00231930436$ with $a_e=(g_s-2)/2$ and Schwinger $\alpha/2\pi$ ; (3) $|\mathbf S|=\tfrac{\sqrt3}{2}\hbar$ ; (4) the spin-flip slope $g_s\mu_B/h=28.02$ GHz/T and X-band $9.81$ GHz at $0.35$ T; (5) the Stern–Gerlach chain ( $v=533$ m/s, $F=1.30\times10^{-20}$ N, split $0.31$ mm). All values are CODATA/textbook constants; none fabricated.

Prerequisites. The Bohr Model of Hydrogen · Solution of the Schrödinger Equation for Hydrogen · Quantum Numbers and Orbital Angular Momentum. Leads to. Fine Structure and Spin-Orbit Coupling · The Zeeman Effect · The Paschen-Back and Anomalous Zeeman Regimes · Hyperfine Structure · Multi-electron Atoms and the Central-Field Approximation.

If nothing is really spinning, why call it angular momentum and why does it make a magnetic moment?

Two separate answers. It is angular momentum because it enters every conservation law as one: atomic transitions balance only when $\hat{\mathbf S}$ is counted alongside $\hat{\mathbf L}$ , and the Einstein–de Haas effect makes it mechanical — remagnetize an iron rod and it visibly rotates. Formally, "angular momentum" means "generator of rotations," and spin rotates the internal state as $\hat{\mathbf L}$ rotates the wavefunction. It makes a moment because a moment is defined by its coupling $\hat H=-\boldsymbol\mu\cdot\mathbf B$ , not by circulating charge; the relativistic wave equation forces that coupling with $g_s=2$ . The current loop is a sufficient mechanism, never a necessary one.

Where does the factor of two in g_s actually come from?

In stages. A current loop and a rigid spinning charged ball both give $g=1$ , so the two is not mechanical. Dirac's equation (1928) produces $g_s=2$ exactly for a structureless spin- $\tfrac12$ particle — relativity's signature, appearing the moment you demand a Lorentz-covariant wave equation. Quantum electrodynamics then dresses the bare electron in virtual photons and pairs, nudging the value up by the anomaly $a_e=(g_s-2)/2\approx1.16\times10^{-3}$ , led by Schwinger's $\alpha/2\pi$ . So $g_s=2$ is relativistic and $g_s-2$ is quantum-field-theoretic — two different layers, not one cosmetic split.

Why is |S| = (√3/2)ħ and not ħ/2 — isn't the spin "one-half"?

The "one-half" is the quantum number $s$ and the maximal projection $S_z=s\hbar=\hbar/2$ , not the length of the vector. The magnitude follows the same rule as any angular momentum, $|\mathbf S|=\sqrt{s(s+1)}\,\hbar$ , which for $s=\tfrac12$ is $\sqrt{3}/2\,\hbar\approx0.866\,\hbar$ — strictly larger than the projection. The gap is the geometric content of the uncertainty relation between spin components: if the vector could lie flat along $z$ then $S_x$ and $S_y$ would both be zero and sharp, which the algebra forbids. So the spin always tilts on a cone, never settling onto the axis.

Silver has 47 electrons — why does the Stern–Gerlach beam behave like a single spin?

Because 46 of them pair off and cancel. Silver's ground configuration closes every shell except a single $5s$ valence electron; paired electrons in a filled subshell contribute zero net spin and zero net orbital angular momentum, so the atom's moment is that of the one unpaired $5s$ electron — an $\ell=0$ state carrying only spin. That is why the two spots are a clean readout of $s=\tfrac12$ and not a many-electron tangle: the atom is a spin- $\tfrac12$ probe in disguise.

The moment is about one Bohr magneton, but the spin is only half a unit — how do both hold?

They hold together precisely because $g_s\approx2$ . The projected moment is $|\mu_{s,z}|=\tfrac{g_s}{2}\mu_B\approx1.001\,\mu_B$ while the angular-momentum projection is $S_z=\tfrac12\hbar$ : half a unit of angular momentum carries very nearly one full magneton of magnetism, the factor of two in $g_s$ closing the books. This near-coincidence ( $g_s/2\approx1$ ) let early workers quote "one Bohr magneton per electron" and still get consistent numbers, which briefly hid the anomaly.

Does the Stern–Gerlach magnet measure the spin, or create the value it reads?

It creates the answer along the axis it defines. Before the magnet the atom is generally in a superposition of up and down along $z$ ; the gradient entangles position with $S_z$ and, once the spot is recorded, collapses the state to one value. Send a purified spin-up beam through a second magnet rotated to $x$ and it splits 50/50 again — the $x$ -measurement scrambles the $z$ -answer, because $\hat S_x$ and $\hat S_z$ do not commute. The two spots are not a pre-existing tilt being revealed; they are the two outcomes of a question the apparatus poses, and its axis is the experimenter's choice.