1. P460 - Spin1 Spin and Magnetic Moments 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 natural unit and the “-” sign is due to the electron’s charge

2. P460 - Spin2 Spin Particles have an intrinsic angular momentum - called spin though nothing is “spinning” probably a more fundamental quantity than mass integer spin --> Bosons half-integer--> Fermions Spin particle postulated particle 0 pion Higgs, selectron 1/2 electron photino (neutralino) 1 photon 3/2 delta 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)

3. P460 - Spin3 Spin 1/2 expectation values similar eigenvalues as orbital angular momentum (but SU(2)) Dirac equation gives g-factor of 2 non-diagonal components (x,y) aren’t zero. Just indeterminate. Can sometimes use Pauli spin matrices to make calculations easier with two eigenstates (eigenspinors)

4. P460 - Spin4 Zeeman Effect Angular momentum->magnetic moment->energy shifts additional terms in S.E. do spin-orbit later. Right now assume atom in external magnetic field look at ground state of H. L=0, S=1/2

5. P460 - Spin5 Spin 1/2 expectation values Let’s assume state in a combination of spin-up and spin-down states (it isn’t polarized). Can calculate some expectation values. Griffiths Ex. 4.2. Z-component x-component as normalized, by inspection

6. P460 - Spin6 Griffiths Prob. 4.28. For the most general normalized spinor find expectation values: just did x and z repeat for other note x and y component will have non-zero “width” for their distributions as not diagonalized

7. P460 - Spin7 Can look at the widths of spin terms if in a given eigenstate z picked as diagonal and so for off-diagonal Widths

8. P460 - Spin8 Assume in a given eigenstate the direction of the total spin can’t be in the same direction as the z-component (also true for l>0) Example: external magnetic field. Added energy puts electron in the +state. There is now a torque which causes a precession about the “z-axis” (defined by the magnetic field) with Larmor frequency of Components, directions, precession B S z

9. P460 - Spin9 Griffiths does a nice derivation of Larmor precession but at the 560 level to understand need to solve problem 4.30. Construct the matrix representing the component of spin angular momentum along an arbitrary radial direction r. Find the eigenvalues and eigenspinors. Put components into Pauli spin matrices and solve for its eigenvalues Angles

10. P460 - Spin10 Go ahead and solve for eigenspinors. Phi phase is arbitrary. gives if r in z,x,y-directions

11. P460 - Spin11 Combining Angular Momentum If have two or more angular momentum, the combination is also eigenstate(s) of angular momentum. Group theory gives the rules: representations of angular momentum have 2 quantum numbers: combining angular momentum A+B+C…gives new states G+H+I….each of which satisfies “2 quantum number and number of states” rules trivial example. Let J= total angular momentm

12. P460 - Spin12 Combining Angular Momentum Non-trivial example. Get maximum J by maximum of L+S. Then all possible combinations of J (going down by 1) to get to minimum value |L-S| number of states when combined equals number in each state “times” each other the final states will be combinations of initial states. The “coefficiants” (how they are made from the initial states) can be fairly easily determined using group theory (and step-up and step-down operaters). Called Clebsch-Gordon coefficients

13. P460 - Spin13 Same example. Example of how states “add”: Note Clebsch-Gordon coefficients 2 terms

14. P460 - Spin14 Clebsch-Gordon coefficients for different J,L,S