The Real Hydrogen Atom 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 (first calculated by Bethe using perturbation theory on train from Long Island to Ithaca)

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 (book does approximation): convert back to lab frame (Thomas precession due to non-inertial frame gives a factor of 2 – Dirac eq gives directly). Energy depends on spin-orbit coupling

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

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 J=3/2, L=1 J=1/2 L=0 J=1/2 L=1 n=2 L=0,1 with relativistic j=3/2 j=1/2

Numerology c2=1/e0m0 have but and so

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 J=3/2 j=1/2 L=1

Relativistic Effects Solved using non-relativistic S.E. can treat relativistic term (Krel) as a perturbation

Relativistic Effects <V> can also use virial theorem by integrating over the radial wave function

Relativistic+spin-orbit Effects combine spin-orbit and relativistic corrections energy levels depend on only n+j (!). Dirac equation gives directly (not as perturbation). For n=2 have:

Energy Levels in Hydrogen Degeneracy = 2j+1 spectroscopic notation: nLj with L=0 S=state, L=1 P-state, L=2 D-state also can note spin “doublet” is single electron with s=1/2 # states N=3 E N=2 N=1

Zeeman Effect:External B Field Energy shift depends on mj and removes any remaining degeneracy. Now two fields (internal and external) and details of splitting depends on relative strengths Unless S=0, the magnetic moment and the total angular momentum are not in the same direction (and aren’t in B direction). For weak external field, manipulating the dot products gives B>0 B=0

Zeeman Effect strong field DL=+-1 Dm=0,+1,-1

Zeeman Effect:External B Field Assume that weak B field (if strong then L and S won’t couple) B field off  1 photon energy B field on  6 photon energies (with their energy depending on the g factor and on the B field One of the first indicators that the electron had intrinsic angular momentum s=1/2 B>0 B=0

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

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)