Magnetic moments and Larmor precession

A magnetic moment is a vector operator, and inside a single spin-half multiplet there is only one vector operator to build it from: Ŝ itself. The projection theorem therefore forces μ̂ = γŜ, and every species-specific fact collapses into the single number γ. Classically a rigid body of charge q and mass m, with charge and mass identically distributed, gives μ = (q/2m)L — a g factor of exactly 1, and orbital motion in an atom does obey gₗ = 1. Spin does not: γ = −gₛ e/2mₑ with gₛ ≈ 2, so an electron is twice as magnetic as its angular momentum classically allows. That γ is an input rather than a derivation shows in the table of values: γₚ/2π = 42.577 MHz T⁻¹ for the proton, while the neutron, with no charge at all, still carries gₙ = −3.826.

Write B = B n̂ and the Zeeman Hamiltonian is Ĥ = −μ̂⋅B = −γB(n̂⋅Ŝ) = (ħω/2)(n̂⋅σ), with ω = −γB. That is a traceless Hermitian 2×2 matrix, so the spectrum is read straight off the eigenvalues ±1 of n̂⋅σ: energies ∓γħB/2 and a splitting ΔE = ħ|γ|B = gₛ μ_B B. The eigenbasis is the pair diagonalised in UPV-10.04 — |n,+⟩ = (cos(θ/2), e^(iφ)sin(θ/2)) and |n,−⟩ = (−sin(θ/2), e^(iφ)cos(θ/2)), two genuinely different spinors and not one formula with a sign buried in it. Watch the ordering: for the electron γ < 0, so E_± = ∓γħB/2 puts |n,+⟩, the state with ⟨S⟩ along B, at the higher energy +|γ|ħB/2. The choice of basis is the boundary condition here, and the field makes it: put B along x and Ŝₓ is what is conserved, while the laboratory's |↑⟩ and |↓⟩ become the superpositions. Numbers then set the technology. An electron at 0.339 T splits by h × 9.5 GHz — X-band microwaves in an ESR cavity. A proton at 1.5 T splits by h × 63.9 MHz — a radio coil. Both gaps are minute beside kT = 25.9 meV at 300 K, and how minute decides the experiment: the equilibrium polarisation tanh(ΔE/2kT) is 7.6 × 10⁻⁴ for that electron but only 5.1 × 10⁻⁶ for that proton, some 150 times smaller, which is why NMR works on a part-in-a-million population difference and lives on signal averaging.

Nothing has to be integrated. Because (n̂⋅σ)² = I, every even power in the series for U(t) = exp(−iĤt/ħ) = exp(−i(ωt/2)(n̂⋅σ)) is I and every odd power is n̂⋅σ, so the series resums to U(t) = cos(ωt/2) I − i sin(ωt/2)(n̂⋅σ). That is exactly the SU(2) rotation operator with rotation angle ωt about n̂: evolving a spinor in a magnetic field is literally rotating it about the field direction, at rate ω = −γB. Two consequences follow from the half-angle. ⟨S⟩ returns after ωt = 2π, but U = −I there, so the state itself needs ωt = 4π; and that minus sign stops being global as soon as the spin is one arm of an interferometer, which is how neutron experiments measured the 4π period directly.

Take B along z, so Ĥ = (ħω/2)σ_z, and start from |ψ(0)⟩ = cos(θ/2)|↑⟩ + sin(θ/2)|↓⟩. Then |ψ(t)⟩ = cos(θ/2)e^(−iωt/2)|↑⟩ + sin(θ/2)e^(+iωt/2)|↓⟩: the moduli are untouched, and only the relative phase runs, at ωt. So P(↑) = cos²(θ/2) and ⟨S_z⟩ = (ħ/2)cos θ are constants of the motion, which is just [Ĥ, Ŝ_z] = 0 restated. What moves is the transverse pair, ⟨Sₓ⟩ = ħ Re(α*β) = (ħ/2) sin θ cos ωt and ⟨S_y⟩ = ħ Im(α*β) = (ħ/2) sin θ sin ωt: fixed length, rotating azimuth. At θ = 60° the state keeps P(↑) = 0.750 for ever while ⟨S⟩ sweeps a cone of transverse radius 0.433ħ. Shifting population needs a field component perpendicular to z; on resonance a rotating B₁ tips the spin at rate γB₁ in the rotating frame, and that is the whole of pulsed NMR.

Every line above assumed B is the same across the wavepacket, so the state factorises as |spatial⟩ ⊗ |spin⟩. Let B depend on position and Ĥ = −γ B(r)⋅Ŝ becomes an entangler: the ↑ and ↓ components pick up different momenta, the packets separate, and the transverse components of ⟨S⟩ die as the overlap ⟨χ₊|χ₋⟩ goes to zero. That is Stern–Gerlach, and it is what measuring S_z physically consists of. The ensemble version is everyday: a spread ΔB across an NMR sample gives a spread of rates |γ|ΔB, and the transverse signal fans out over T₂* ≈ 1/(πΔf). Protons at 1.5 T with ΔB = 1.0 μT give Δf = 42.6 Hz and T₂* ≈ 7.5 ms — some 4.8 × 10⁵ precessions before the free-induction decay is gone. Because that dephasing is static, a π pulse reverses it and rebuilds an echo; only genuine T₂ processes survive.

The Dirac equation for a point spin-half in a field predicts gₛ = 2 exactly — the first real explanation of the factor Uhlenbeck and Goudsmit had to insert by hand, and a decisive argument against any picture of the electron as spinning charged matter. Schwinger's one-loop correction adds aₑ = (gₛ − 2)/2 = α/2π = 1.1614 × 10⁻³; carried further, aₑ = α/2π − 0.3285(α/π)² + … = 1.15965 × 10⁻³, agreeing with experiment to about a part in 10¹². The measurement is exactly the physics of this topic: compare the spin-precession frequency with the cyclotron frequency of the same electron in the same field. Composite particles sit nowhere near 2 — gₚ = 5.586, gₙ = −3.826 — and the muon anomaly, measured the same way at Fermilab, is watched because a discrepancy there would be new physics.