1. P460 - spin-orbit1 Energy Levels in Hydrogen Solve SE and in first order get (independent of L): can use perturbation theory to determine: magnetic effects (spin-orbit and hyperfine e-A) relativistic corrections Also have Lamb shift due to electron “self- interaction”. Need QED (Dirac eq.) and depends on H wavefunction at r=0 (source of electric field). Very small and skip in this course

2. P460 - spin-orbit2 Spin-Orbit Interactions A non-zero orbital angular momentum L produces a magnetic field electron sees it. Its magnetic moment interacts giving energy shift in rest frame of electron, B field is (see book): convert back to lab frame (Thomas precession due to non-inertial frame gives a factor of 2). Energy depends on spin-orbit coupling

3. P460 - spin-orbit3 Spin-Orbit: Quantum Numbers The spin-orbit coupling (L*S) causes m l and m s to no longer be “good” quantum numbers spin-orbit interactions changes energy. In atomic physics, small perturbation, and can still use H spatial and spin wave function as very good starting point. Large effects in nuclear physics (and will see energy ordering very different due to couplings)

4. P460 - spin-orbit4 SL Expectation value Determine expectation value of the spin-orbit interaction using perturbation theory. Assume J,L,S are all “good” quantum numbers (which isn’t true) assume H wave function is ~eigenfunction of perturbed potential

5. P460 - spin-orbit5 SL Expectation value To determine the energy shift, also need the expectation value of the radial terms using Laguerre polynomials put all the terms together to get spin-orbit energy shift. =0 if l=0 L=1 J=3/2 j=1/2

6. P460 - spin-orbit6 Spin Orbit energy shift For 2P state. N=2, L=1, J= 3/2 or 1/2 and so energy split between 2 levels is L=1 J=3/2 j=1/2

7. P460 - spin-orbit7 Relativistic Effects Solved using non-relativistic S.E. can treat relativistic term (Krel) as a perturbation can use virial theorom

8. P460 - spin-orbit8 Relativistic+spin-orbit Effects by integrating over the radial wave function combine spin-orbit and relativistic corrections energy levels depend on only n+j (!). Dirac equation gives directly (not as perturbation)

9. P460 - spin-orbit9 Energy Levels in Hydrogen Degeneracy = 2j+1 spectroscopic notation: nL j with L=0 S=state, L=1 P-state, L=2 D-state E N=1 N=3 N=2 # states

10. P460 - spin-orbit10 Hyperfine Splitting Many nuclei also have spin p,n have S=1/2. Made from 3 S=1/2 quarks (plus additional quarks and antiquarks and gluons). G- factors are 5.58 and -3.8 from this (-2 for electron). Nuclear g-factors/magnetic moments complicated. Usually just use experimental number for Hydrogen. Let I be the nuclear spin (1/2) have added terms to energy. For S-states, L=0 and can ignore that term

11. P460 - spin-orbit11 Hyperfine Splitting Electron spin couples to nuclear spin so energy difference between spins opposite and aligned. Gives 21 cm line for hydrogen (and is basis of NMR/MRI)