P460 - Spin1 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 - 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)

P460 - Spin3 Spin 1/2 expectation values similar eigenvalues as orbital angular momentum (but SU(2)). No 3D “function” 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)

P460 - Spin4 Spin 1/2 expectation values “total” spin direction not aligned with any component. can get angle of spin with a component

P460 - Spin5 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 - Spin6 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 - Spin7 Can also determine and widths

P460 - Spin8 Can look at the widths of spin terms if in a given eigenstate z picked as diagonal and so for off-diagonal Widths- example

P460 - Spin9 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

P460 - Spin10 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 Precession - details

P460 - Spin11 Assume at t=0 in the + eigenstate of S x Solve for the x and y expectation values. See how they precess around the z-axis Precession - details

P460 - Spin12 can look at any direction (p 160 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 Arbitrary Angles

P460 - Spin13 Go ahead and solve for eigenspinors. Phi phase is arbitrary. gives if r in z,x,y-directions

P460 - Spin14 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 - Spin15 Combining Angular Momentum Non-trivial examples. add 2 spins add spin and orbital angular momentum

P460 - Spin16 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 - Spin17 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 S 2 step down from ++ state Clebsch-Gordon coefficients

P460 - Spin18 Combining Ang. Momentum check that eigenstates have right eigenvalue for S 2 first write down and then look at terms putting it all together see eigenstates

P460 - Spin19 L=1 + S=1/2 Example of how states “add”: Note Clebsch-Gordon coefficients 2 terms

P460 - Spin20 Clebsch-Gordon coefficients for different J,L,S