Spin and Magnetic Moments (skip sect. 10-3)

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 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  Bosons half-integer Fermions • Spin particle postulated particle • 0 pion Higgs, selectron • 1/2 electron photino (neutralino) • 1 photon • 3/2 D • 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) 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 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) P460 - Spin

5. Spin 1/2 expectation values • “total” spin direction not aligned with any component. • can get angle of spin with a component 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 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 • or could rotate wavefunction to basis where x is diagonal P460 - Spin

8. Can also determine • and widths P460 - Spin

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

10. Components, directions, precession • 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 z B S P460 - Spin

11. Precession - details • Hamiltonian for an electron in a magnetic field • assume solution of form • If B direction defines z-axis have Scr.eq. • And can get eigenvalues and eigenfunctions P460 - Spin

12. Precession - details • Assume at t=0 in the + eigenstate of Sx • Solve for the x and y expectation values. See how they precess around the z-axis P460 - Spin

13. Arbitrary Angles • can look at any direction (p 160 and problem 10-2 or see Griffiths 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 P460 - Spin

14. Go ahead and solve for eigenspinors. • Gives (phi phase is arbitrary) • if r in z,x,y-directions P460 - Spin

15. Combining Angular Momentum • If have two or more angular momentum, the combination is also an 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 momentum P460 - Spin

16. Combining Angular Momentum • Non-trivial examples. add 2 spins. The z-components add “linearly” and the total ads “vectorally”. Really means add up z-component and then divide up states into SU(2) groups 4 terms. need to split up. The two 0 mix P460 - Spin

17. Combining Angular Momentum • add spin and orbital angular momentum P460 - Spin

18. Combining Angular Momentum • 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 “coefficients” (how they are made from the initial states) can be fairly easily determined using group theory (step-down operaters). Called Clebsch-Gordon coefficients • these give the “dot product” or rotation between the total and the individual terms. P460 - Spin

19. Combining Angular Momentum • example 2 spin 1/2 • have 4 states with eigenvalues 1,0,0,-1. Two 0 states mix to form eigenstates of S2 • step down from ++ state • Clebsch-Gordon coefficients P460 - Spin

20. Combining Ang. Momentum • check that eigenstates have right eigenvalue for S2 • first write down • and then look at terms • putting it all together see eigenstates P460 - Spin

21. 2 terms • L=1 + S=1/2 • Example of how states “add”: • Note Clebsch-Gordon coefficients (used in PHYS 374 class for Mossbauer spectroscopy) P460 - Spin

22. Clebsch-Gordon coefficients for different J,L,S P460 - Spin