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# 08spin - Spin and Magnetic Moments(skip sect 10-3 •...

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P460 - Spin 1 Spin and Magnetic Moments (skip sect. 10-3) Orbital and intrinsic (spin) angular momentum produce magnetic moments coupling between moments shift atomic energies · Look first at orbital (think of current in a loop) · the “g-factor” is 1 for orbital moments. The Bohr magneton is introduced as the natural unit and the “-” sign is due to the electron’s charge L g L mvr L but r area current A I b l l m q r qv r r h μ π μ π μ = = = = × = = 2 2 2 e b m e 2 h = μ l b l zl b l l m g l l g l l L μ μ μ μ = + = + = ) 1 ( ) 1 ( 2 h

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P460 - Spin 2 Spin Particles have an intrinsic angular momentum - called spin though nothing is “spinning” probably a more fundamental quantity than mass integer spin barb2right Bosons half-integer barb2right Fermions Spin particle postulated particle 0 pion Higgs, selectron 1/2 electron photino (neutralino) 1 photon 3/2 Δ 2 graviton relativistic QM uses Klein-Gordon and Dirac equations for spin 0 and 1/2. Solve by substituting operators for E,p. The Dirac equation ends up with magnetic moment terms and an extra degree of freedom (the spin) 2 2 2 2 2 : : m p E D m p E KG + = + =
P460 - Spin 3 Spin 1/2 expectation values similar eigenvalues as orbital angular momentum (but SU(2)). No 3D “function” Dirac equation gives g-factor of 2 [ ] 2 00232 . 2 , , | | ... | | , ) 1 ( , 2 1 2 1 2 4 3 2 2 3 2 1 2 2 1 2 2 = = = = = = + = s S g s z z k ijk j i g S S S for s s S s s S S S S b s h r r h h h h h h μ μ ε

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P460 - Spin 4 Spin 1/2 expectation values non-diagonal components (x,y) aren’t zero. Just indeterminate. Can sometimes use Pauli spin matrices to make calculations easier with two eigenstates (eigenspinors) = = = = = 0 0 0 1 1 0 1 0 0 1 1 0 0 1 2 2 2 2 4 3 2 2 i i S S S S S y x z i i h h h h h σ 2 2 1 0 0 1 h h = = + = = + + eigenvalue S eigenvalue S z z χ χ
P460 - Spin 5 Spin 1/2 expectation values “total” spin direction not aligned with any component. can get angle of spin with a component 3 1 4 3 2 1 cos = = = h h r S S z β

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P460 - Spin 6 Spin 1/2 expectation values Let’s assume state in an arbitrary combination of spin-up and spin-down states. expectation values. z-component x-component y-component 1 | | | | 2 2 = + = + = + b a with b a b a χ χ χ ( ) ) ( 1 0 0 1 | | 2 2 2 2 * * b a b a b a S S z z = = = h h χ χ ( ) ) ( 0 0 * * 2 2 2 * * a b b a b a b a S x + = = h h h ( ) ) ( 0 0 * * 2 2 2 * * a b b a i b a i i b a S y = = h h h
P460 - Spin 7 Spin 1/2 expectation values example assume wavefunction is expectation values. z-component x-component Can also ask what is the probability to have different components. As normalized, by inspection

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08spin - Spin and Magnetic Moments(skip sect 10-3 •...

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