08spin - P460 - Spin 1 Spin and Magnetic Moments (skip...

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Unformatted text preview: 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 electrons charge L g L mvr L but r area current A I b l l m q r qv r r h = = = = = = 2 2 2 e b m e 2 h = l b l zl b l l m g l l g l l L = + = + = ) 1 ( ) 1 ( 2 h 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 b Bosons half-integer b Fermions Spin particle postulated particle 0 pion Higgs, selectron 1/2 electron photino (neutralino) 1 photon 3/2 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 P460 - Spin 4 Spin 1/2 expectation values non-diagonal components (x,y) arent zero. Just indeterminate. Can sometimes use Pauli spin matrices to make calculations easier with two eigenstates (eigenspinors) = = = = = 1 1 1 1 1 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 2 2 1 1 h h = = + = = + + eigenvalue S eigenvalue S z z 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 P460 - Spin 6 Spin 1/2 expectation values Lets 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 ( ) ) ( 1 1 | | 2 2 2 2 * * b a b a b a S S z z = = = h h ( ) ) ( * * 2 2 2 * * a b b a b a b a S x + = = h h h ( ) ) ( * * 2 2 2 * * a b b a i b a i i b a S y =...
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This note was uploaded on 10/02/2009 for the course PHYS 460 taught by Professor Johnson,c during the Spring '08 term at Northern Illinois University.

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08spin - P460 - Spin 1 Spin and Magnetic Moments (skip...

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