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

Spin Spin particle postulated particle 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 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

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

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

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

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

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

Can also determine and widths P460 - Spin

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

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 S B z P460 - Spin

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

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

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

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

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

Combining Angular Momentum Non-trivial examples. add 2 spins. The z-components add “linearly” and the total adds “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

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

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

Combining Angular Momentum Clebsch-Gordon coefficients these give the “dot product” or rotation between the total and the individual terms. “easy” but need to remember what different quantum number labels refer to P460 - Spin

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

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

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

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