Reminder: Spin precession Spin part of Hamiltonian: Hspin = −µ·B = −γ S·B γ = gq /2m (gyromagnetic ratio) Acts on “spin space” only, so really: Hspin = − γ Iwave fn  S·B So we ignore spatial wave function For B in z direction, say Hspin = ω0Sz ω0 = γ B Time-independent Hamiltonian, so solution is

Matrix Representation If starting state is spin-left, i.e. a0 = b0 = 1/√2, then we get precession with period 2π/ω0: NB after one period, overall sign reversed.

Neutron interferometry ATOMINSTITUT Vienna Beam split by Bragg diffraction in vertical crystal planes Whole interferometer carved from a perfect crystal of silicon: Separation between elements exact multiple of inter-atomic spacing. Only ~ a dozen successfully made in 30 years! Credit: NIST, Boulder, Colorado

2π rotation (Warner et al 1975) g factor of neutron = −3.83 i.e.  = ge/2mn even though no net charge! Precession frequency ω0=−geB/2mn de Broglie λ = 0.1445 nm Effective ℓ ≈ 2.7 cm allowing for leakage of B outside magnet. Time in field t = ℓ/v = ℓmnλ/h Angle precessed: ω0t = −geBλℓ/2h For 2π rotation we need B = 3.4 mT = 34 gauss Observed period ≈ 62 G 4π rotation needed to restore original state: 2π rotation changes sign, as predicted by QM.