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Reminder: Spin precession

Reminder: Spin precession. Spin part of Hamiltonian: H spin = − µ·B = − γ S·B γ = gq /2 m ( gyromagnetic ratio ) Acts on “spin space” only, so really: H spin = − γ I wave fn  S·B So we ignore spatial wave function For B in z direction, say H spin = ω 0 S z

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Reminder: Spin precession

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  1. Reminder: Spin precession • Spin part of Hamiltonian: • Hspin = −µ·B = −γS·B • γ = gq/2m (gyromagnetic ratio) • Acts on “spin space” only, so really: • Hspin = − γIwave 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

  2. 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.

  3. 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! Neutron interferometry ATOMINSTITUT Vienna Credit: NIST, Boulder, Colorado

  4. 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. 2π rotation (Warner et al 1975)

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